Elements of philosophy the first section, concerning body / written in Latine by Thomas Hobbes of Malmesbury ; and now translated into English ; to which are added Six lessons to the professors of mathematicks of the Institution of Sr. Henry Savile, in the University of Oxford.

OF THE PROPORTIONS OF MOTIONS AND MAGNITVDES.

CHAP. XV. Of the Nature, Properties, and diverse Con∣siderations of Motion and Endeavour.

• 1 Repetition of some Principles of the doctrine of Motion formerly set down.
• 2 Other Principles added to them.
• 3 Certain Theoremes concerning the nature of Motion.
• 4 Diverse Considerations of Motion▪
• 5 The way by which the first Endeavour of Bodies Moved ••endoth.
• 6 In Motion which is made by Concourse, one of the Movents ceasing, the Endeavour is made by the way by which the rest tend.
• 7 The Endeavour of any Moved Body, which having its Motion in the Circumference of a Circle, parts from the same, proceeds afterwards in a straight line which toucheth the Circle.
• 8 How much greater, the Velocity or Magnitude is of a Movent, so much greater is the Efficacy thereof upon any other Body in its way.

1 THe next things in order to be treat∣ed of are MOTION and MAG∣NITUDE, which are the most common Accidents of all Bodies. This place therefore most proper∣ly belongs to the Elements of Ge∣ometry. But because this part of Philosophy having been improved by the best Wits of all Ages has afforded greater plenty of matter then can well be thrust together within the narrow limits of this discourse; I thought fit to admonish the Reader, that before he

proceed further, he take into his hands the Works of Euclide, Archimedes, Apollonius and other as well Ancient as Modern Wri∣ters. For to what end is it to do over again that which is already done? The little therefore that I shall say concerning Geometry in some of the following Chapters, shall be such onely as is new, and conducing to Natural Philosophy.

I have already delivered some of the Principles of this doctrine in the 8 & 9 Chapters, which I shall briefly put together here, that the Reader in going on may have their light neerer at hand.

First therefore in the 8th Chap. and 10 Article, Motion is defined to be the continual privation of one place, and acquisition of another.

Secondly, it is there shewn, that Whatsoever is Moved is Moved in Time.

Thirdly, in the same Chap. 11. Article, I have defined Rest to be when a Body remains for some time in one place.

Fourthly, it is there shewn, that Whatsoever is Moved is not in any determined place; as also that the same has been Moved, is still Moved, and will yet be Moved; So that in every part of that Space in which Motion is made, we may consider three Times, namely the Past, the Present, and the Future Time.

Fiftly, in the 15 Article of the same Chapter, I have defined Ve∣locity or Swiftness to be Motion considered as Power, namely, that Power by which a Body Moved may in a certain Time transmit a certain Length; which also may more briefly be enunciated thus, Velocity is the quantity of Motion determined by Time and Line.

Sixthly, in the same Chap. 16. Article, I have shewn that Motion is the Measure of Time.

Seventhly, in the same Chap. 17th Art. I have defined Motions to be Equally Swift, when in Equal Times Equal Lengths are transmitted by them.

Eighthly, in the 18 Article of the same Chapter, Motions are de∣fined to be Equal, when the Swiftness of one Moved Body computed in e∣very part of its magnitude, is equal to the Swiftness of another computed also in every part of its magnitude. From whence it is to be noted, that Motions Equal to one another, and Motions Equally Swift, do not sig∣nifie the same thing; for when two horses draw abrest, the Moti∣on of both is greater then the Motion of either of them singly; but the Swiftness of both together is but Equal to that of either.

Ninthly, in the 19 Article of the same Chapter, I have shewn,

that Whatsoever is at Rest will alwayes be at Rest, unless there be some other Body besides it, which by getting into its place, suffers it no longer to remain at Rest. And that Whatsoever is Moved, will alwayes be Moved, unless there be some other Body besides it, which hinders its Motion.

Tenthly, In the 9 Chapter and 7 Article, I have demonstrated, that When any Body is moved which was formerly at Rest, the immediate ef∣ficient cause of that Motion is in some other Moved and Contiguou•• Body.

Eleventhly, I have shewn in the same place, that Whatsoever is Moved, will always be Moved in the same way, and with the same Swift∣ness, if it be not hindered by some other Moved and Contiguou•• Body.

2 To which Principles I shall here add these that follow. First, I define ENDEAVOUR to be Motion made in less Space and Time then can be given; that is, less then can be determined or assigned by Exposition or Number; that is, Motion made through the length of a Point, and in an Instant or Point of Time. For the explayning of which Definition it must be remembred, that by a Point is not to be un∣derstood that which has no quantity, or which cannot by any means be divided (for there is no such thing in Nature); but that whose quantity is not at all considered, that is, whereof neither quantity nor any part is computed in demonstration; so that a Point is not to be taken for an Indivisible, but for an Undivided thing; as also an Instant is to be taken for an Undivided, and not for an Indivisible Time.

In like manner Endeavour is to be conceived as Motion; but so, as that neither the quantity of the Time in which, nor of the Line in which it is made may in demonstration be at all brought into comparison with the quantity of that Time, or of that Line of which it is a part. And yet, as a Point may be compared with a Point, so one Endeavour may be compared with another Endea∣vour, and one may be found to be greater or lesse then another. For if the Vertical points of two Angles be compared, they will be equal or unequal in the same proportion which the Angles them∣selves have to one another. Or if a straight Line cut many Cir∣cumferences of Concentrick Circles, the inequality of the points of intersection will be in the same proportion which the Perime∣ters have to one another. And in the same manner, if two Moti∣ons begin and end both together, their Endeavours will be Equal or Unequal according to the proportion of their Velocities; as we

see a bullet of Lead descend with greater Endeavour then a ball of Wooll,

Secondly, I define IMPETUS or Quickness of Motion, to be the Swiftness or Velocity of the Body moved, but considered in the several points of that time in which it is moved; In which sense Impetus is nothing else but the quantity or velocity of Endeavour. But considered with the whole time, it is the whole velocity of the Body moved, taken together throughout all the time, and equal to the Product of a Line representing the time multiplyed into a Line representing the arithmetically mean Impetus or Quickness. Which Arithmetical Mean what it is, is defined in the 29th Arti∣cle of the 13th Chapter.

And because in equal times the wayes that are passed are as the Velocities, and the Impetus is the Velocity they go withal reckoned in all the several points of the times, it followeth that during any time whatsoever, howsoever the Impetus be encreased or decrea∣sed, the length of the way passed over shall be encreased or de∣creased in the same proportion; and the same Line shall repre∣sent both the way of the Body moved, and the several Impetus or degrees of Swiftness wherewith the way is passed over.

And if the Body moved be not a point, but a straight line mo∣ved so as that every point thereof make a several straight line, the Plain described by its motion, whether Uniform, Accelerated or Retarded, shall be greater or less (the time being the same) in the same proportion with that of the Impetus reckoned in one motion to the Impetus reckoned in the other. For the reason is the same in Parallelograms and their Sides,

For the same cause also if the Body moved be a Plain, the Solid described shall be still greater or less in the proportions of the se∣veral Impetus or Quicknesses reckoned through one Line, to the se∣veral Impetus reckoned through another.

This understood, let ABCD (in the first figure of the 17th Chapter) be a Parallelogram; in which suppose the side AB to be moved parallelly to the opposite side CD, decreasing al the way till it vanish in the point C, and so describing the figure ABEFC; the point B as AB decreaseth, will therefore describe the Line BEFC; and suppose the time of this motion designed by the line CD; and in the same time CD suppose the side AC to be moved parallelly and uniformly to BD. From the point O taken at adven∣ture

in the Line CD, draw OR parallel to BD, cutting the Line BEFC in E, and the side AB in R. And again from the point Q taken also at adventure in the Line CD, draw QS parallel to BD, cutting the Line BEFC in F, and the side AB in S; and draw EG and FH parallel to CD, cutting AC in G and H. Lastly, suppose the same construction done in all the points possible of the Line BEFC. I sa••, that as the proportions of the Swiftnesses wherewith QF, OE, DB, and all the rest supopsed to be drawn parallel to DB, and terminated in the Line BEFC, are to the pro∣portions of their several Times designed by the several parallels HF, GE, AB and all the rest supposed to be drawn parallel to the Line of time CD, and terminated in the Line BEFC (the ag∣gregate to the aggregate) so is the Area or Plain DBEFC to the Area or Plain ACFEB. For as AB decreasing continually by the line BEFC vanisheth in the time CD into the point C, so in the same time the line DC continually decreasing vanisheth by the same line CFEB into the point B; and the point D de∣scribeth in that decreasing motion the line DB equall to the line AC described by the point A in the decreasing motion of A & B; & their swiftnesses are therefore equal. Again, because in the time GE the point O describeth the line OE, and in the same time the point R describeth the line RE, the line OE shall be to the line RE, as the swiftness wherewith OE is described to the swiftness wherwith RE is described. In like māner, because in the same time HF the point Q describeth the Line QF, and the point S the Line SF, it shall be as the swiftness by which QF is described to the swiftness by which SF is described, so the Line it self QF to the Line it self SF; and so in all the Lines that can possibly be drawn parallel to BD in the points where they cut the Line BEFC. But all the parallels to BD, as RE, SF, AC and the rest that can possi∣bly be drawn from the Line AB to the Line BEFC make the Area of the Plain ABEFC; and all the parallels to the same BD, as QF, OE, DB & the rest drawn to the points where they cut the same Line BEFC make the Area of the Plain BEFCD. As there∣fore the aggregate of the Swiftnesses wherwith the Plain BEFCD is described is to the aggregate of the Swiftnesses wherewith the Plain ACFEB is described, so is the Plain it self BEFCD to the

Plain it self ACFEB. But the aggregate of the Times represented by the parallels AB, GE, HF and the rest, maketh also the Area ACFEB. And therefore as the aggregate of all the Lines QF, OE, DB and all the rest of the Lines parallel to BD and termina∣ted in the Line BEFC is to the aggregate of all the Lines HF, GE, AB and all the rest of the Lines parallel to CD & terminated in the same Line BEFC; that is, as the aggregate of the Lines of Swiftness to the aggregate of the lines of Time, or as the whole Swiftness in the parallels to DB to the whole Time in the parallels to CD, so is the Plain BEFCD to the Plain ACFEB. And the proportions of QF to FH, and of OE to EG, and of DB to BA, and so of all the rest taken together, are the proportion of the Plain DBEFC to the Plain ABEFC. But the Lines QF, OE, DB and the rest are the Lines that designe the Swifness; and the Lines HF, GE, AB & the rest are the Lines that designe the Times of the motions; and therefore the proportion of the Plain DBEFC to the Plain ABEFC is the proportions of all the Velocities taken together, to all the Times taken together. Wherefore as the proportions of the Swiftnesses, &c. which was to be demonstrated.

The same holds also in the diminution of the Circles whereof the lines of Time are the Semidiameters, as may easily be concei∣ved by imagining the whole Plain ABCD turned round upon the Axis BD; for the Line BEFC will be every where in the Super∣ficies so made, and the Lines HG, GE, AB which here are Paral∣lelograms will be there Cylinders, the Diameters of whose bases are the lines HF, GE, AB, &c. and the Altitude a point, that is to say, a quantity less then any quantity that can possibly be named; and the Lines QF, OE, DB, &c. small solids whose lengths and breadths are less then any quantity that can be named.

But this is to be noted, that unless the proportion of the summe of the Swiftnesses to the proportion of the summe of the Times be determined, the proportion of the Figure DBEFC to the Figure ABEFC cannot be determined.

Thirdly, I define RESISTANCE to be the endeavour of one moved Body, either wholly or in part contrary to the endeavour of another moved Body, which toucheth the same. I say wholly contrary, when the endeavour of two Bodies proceeds in the same straight Line from the opposite extremes, and contrary in part, when two Bodies

have their endeavour in two Lines, which proceeding from the extreme points of a straight Line, meet without the same.

Fourthly, that I may define what it is to PRESSE, I say that Of two moved Bodies one Presses the other, when with its Endeavour it makes either all or part of the other Body to go out of its place.

Fifthly, A Body which is pressed and not wholly removed is said to RESTORE it self, when (the pressing Body being taken away) the parts which were moved, do by reason of the internal constitution of the pressed Body, return every one into its own place. And this we may observe in Springs, in blown Bladders, and in many other Bodies, whose parts yeild more or less to the Endeavour which the pressing Body makes at the first arrival; but afterwards (when the pressing Body is removed) they do by some force within them Restore themselves, and give their whole Body the same figure it had before.

Sixthly, I define FORCE to be the Impetus or Quickness of Mo∣tion multiplyed either into it self, or into the Magnitude of the Movent, by means wherof the said Movent works more or less upon the Body that resists it.

3 Having premised thus much, I shal now demonstrate, First, That if a point moved come to touch another point which is at rest, how little soever the Impetus or quickness of its motion be, it shall move that other point. For if by that Impetus it do not at all move it out of its place, neither shall it move it, with double the same Im∣petus; for nothing doubled is still nothing; and for the same rea∣son it shall never move it with that Impetus how many times soever it be multiplyed, because nothing how soever it be multiplyed will for ever be nothing. Wherefore when a point is at rest, if it do not yeild to the least Impetus, it will yeild to none, and consequently it will be impossible that that which is at rest should ever be moved.

Secondly, that when a point moved, how little soever the Impetus thereof be, falls upon a point of any Body at rest, how hard soever that Body be, it will at the first touch make it yeild a little. For if it do not yeild to the Impetus which is in that point, neither will it yeild to the Impetus of never so ma∣ny points, which have all their Impetus severally equal to the Im∣petus of that point. For seeing all those points together work equal∣ly, if any one of them have no effect, the aggregate of them all together shall have no effect as many times told as there are points in the whole Body, that is, still no effect at all; and by con∣sequent

there would be some Bodies so hard that it would be im∣possible to break them; that is, a finite hardnesse, or a finite force would not yeild to that which is infinite; which is absurd.

Corollary. It is therefore manifest, that Rest does nothing at all, nor is of any efficacy; and that nothing but Motion gives Motion to such things as be at Rest, and takes it from things moved,

Thirdly, that Cessation in the Movent does not cause Cessation in that which was moved by it. For (by the 11th Number of the 1 Article of this Chapter) whatsoever is moved, persevers in the same way, & with the same Swiftness, as long as it is not hindered by some thing that is moved against it. Now it is manifest, that Cessation is not contrary Motion; and therefore it follows, that the standing still of the Movent, does not make it necessary that the thing moved should also stand still.

Corollary. They are therefore deceived, that reckon the taking away of the impediment or resistance, for one of the causes of Motion.

4 Motion is brought into account for divers respects; First, as in a Body Undivided, (that is, considered as a point); or, as in a Divided Body. In an Undivided Body, when we suppose the way by which the Motion is made, to be a Line; and in a Divided Bo∣dy, when we compute the Motion of the several parts of that Bo∣dy, as of Parts.

Secondly, From the diversity of the regulation of Motion, it is in a Body considered as Undivided, sometimes Uniform, and some∣times Multiform. Uniform is that by which equal Lines are alwayes transmitted in equal times; & Multiform, when in one time more, in another time less space is transmitted. Again, of Multiform Mo∣tions, there are some in which the degrees of Acceleration and Retardation proceed in the same proportions which the Spaces transmitted have, whether duplicate, or triplicate, or by what∣soever number multiplyed; and others in which it is otherwise.

Thirdly, from the number of the Movents; that is, one Motion is made by one Movent onely, and another by the concourse of many Movents.

Fourthly, from the position of that Line in which a Body is mo∣ved, in respect of some other Line; and from hence one Motion is called Perpendicular, another Oblique, another Parallel.

Fifthly, from the position of the Movent in respect of the Moved Body; from whence one Motion is Pulsion or Driving; another Tracti∣on or Drawing. PULSION, when the Movent makes the Moved Body goe before it; and TRACTION, when it makes it follow. Again, there are two sorts of Pulsion; one, when the motions of the Movent and Moved Body begin both together, which may be cal∣led TRUSION or Thrusting and VECTION; the other, when the Movent is first moved, and afterwards the Moved Body, which Motion is called PERCUSSION or Stroke.

Sixthly, Motion is considered sometimes from the Effect onely which the Movent works in the Moved Body, which is usually cal∣led Moment. Now MOMENT is the Excess of Motion which the Movent has, above the Motion or Endeavour of the Resisting Body.

Seventhly, it may be considered from the diversity of the Me∣dium; as one Motion may be made in Vacuity or empty Place; another in a fluid; another in a consistent Medium, that is, a Medium whose parts are by some power so consistent and cohering, that no part of the same will yeild to the Movent, unless the whole yeild also.

Eighthly, when a Moved Body is considered as having parts, there arises another distinction of Motion into Simple and Compoun∣ded. Simple, when all the several parts describe several equal lines; Compounded, when the lines described are Unequal.

5 All Endeavour tends towards that part, that is to say, in that way which is determined by the Motion of the Movent, if the Mo∣vent be but one; or, if there be many Movents, in that way which their concourse determines. For example, if a Moved Body have direct Motion, its first Endeavour will be in a Straight line; if it have Circular Motion, its first Endeavour will be in the Circumfe∣rence of a Circle; & whatsoever the line be in which a Body has its Motion from the concourse of two Movents, as soon as in any point thereof the force of one of the Movents ceases, there immediate∣ly the former Endeavour of that Body will be changed into an En∣deavour in the line of the other Movent.

6 Wherefore, when any Body is carried on by the concourse of two Winds, one of those Winds ceasing, the Endeavour and Mo∣tion of that Body will be in that line, in which it would have been carried by that Wind alone which blows still. And in the de∣scribing

of a Circle, where that which is moved has its Motion determined by a Movent in a Tangent, and by the Radius which keeps it in a certain distance from the Center, if the retention of the Radius cease, that Endeavour which was in the Circum∣ference of the Circle, will now be in the Tangent, that is, in a Straight line. For seeing Endeavour is computed in a lesse part of the Circumference then can be given, that is, in a point, the way by which a Body is moved in the Circumference is com∣pounded of innumerable Straight lines; of which every one is less then can be given, which are therefore called Points. Wherefore when any Body which is moved in the Circumference of a Cir∣cle, is freed from the retention of the Radius, it will proceed in one of those Straight lines, that is, in a Tangent.

7 All Endeavour, whether strong or weak, is propagated to in∣finite distance; for it is Motion. If therefore the first Endeavour of a Body be made in Space which is empty, it will alwayes pro∣ceed with the same Velocity; for it cannot be supposed that it can receive any resistance at all from empty Space; and therefore (by the 7 Article of the 9 Chapter) it will alwayes proceed in the same way and with the same Swiftness. And if its Endeavour be in Space which is filled, yet seeing Endeavour is Motion, that which stands next in its way shall be removed, and endeavour further, and again remove that which stands next, & so infinitely. Where∣fore the propagation of Endeavour from one part of full Space to another, proceeds infinitely. Besides, it reaches in any instant to a∣ny distance, how great soever; For in the same instant in which the first part of the full Medium removes that which is next it, the second also removes that part which is next to it; and therefore all Endeavour, whether it be in empty or in full Space, proceeds not onely to any distance how great soever, but also in any time how little soever, that is, in an instant. Nor makes it any matter, that Endeavour by proceeding growes weaker and weaker, till at last it can no longer be perceived by Sense; for Mo∣tion may be insensible; and I do not here examine things by Sense and Experience, but by Reason.

8 When two Movents are of equal Magnitude, the swifter of them works with greater force then the slower upon a Body that

resists their Motion. Also if two Movents have equal Velocity, the greater of them works with more force then the less. For where the Magnitude is equal, the Movent of greater Velocity makes the greater impression upon that Body upon which it falls; and where the Velocity is equal, the Movent of greater Magnitude falling upon the same point, or an equal part of another Body, lo∣ses less of its Velocity, because the resisting Body works onely up∣on that part of the Movent which it touches, and therefore abates the Impetus of that part onely, whereas in the mean time the parts which are not touched proceed, and retein their whole force till they also come to be touched, and their force has some effect. Wherfore (for example) in Batteries, a longer then a shorter piece of Timber of the same thickness and velocity, and a thicker then a slenderer piece of the same length and velocity, works a greater effect upon the Wall.

CHAP. XVI. Of Motion Accelerated and Vniform, and of Motion by Concourse.

• 1 The Velocity of any Body, in what Time soever it be computed, is that which is made of the multiplication of the Impetus, or Quickness of its Mo∣tion into the Time. 2, &c. In all Motion, the Lengths which are passed through, are to one another, as the Products made by the Impetus multipyed into the Time.
• 6 If two Bodies be moved with Uniform Motion through two Lengths, the proportion of those Lengths to one another will be compounded of the proportions of Time to Time, and Impetus to Impetus, directly taken.
• 7 If two Bodies pass through two Lengths with Uniform Motion, the propor∣tion of their Times to one another will be compounded of the proportions of Length to Length and Impetus to Impetus, reciprocally taken; also the propor∣tion of their Impetus to one another will be compounded of the proportions of Length to Length and Time to Time, reciprocally taken.
• 8 If a Body be carried on with Uniform Motion by two Movents together, which meet in an Angle, the line by which it passes will be a straight line subtending the complement of that Angle to two right Angles. 9, &c. If a Body be carried by two Movents together, one of them being moved with Uniform, the other with Accelerated Motion, and the proportion of their Lengths to their Times be∣ing explicable in numbers, How to find out what line that Body describes.

1 THe Velocity of any Body, in whatsoever Time it be moved, has its quantity determined by the sum of all the several Quicknesses or Impetus which it hath in the several points of the Time of the Bodies Motion. For seeing Velocity (by the Definition of it Chap. 8. Art. 15.) is that Power by which a Body can in a certain time pass through a certain length; and Quickness of Motion, or Impetus (by the 15 Chap. Artic. 2. Numb. 2.) is Velocity taken in one point of time onely, all the Impetus to∣gether taken in all the points of time, will be the same thing with the Mean Impetus multiplyed into the whole Time, or which is all one, will be the Velocity of the whole Motion.

Corollary. If the Impetus be the same in every point, any straight line representing it may be taken for the measure of Time; and the Quicknesses or Impetus applyed ordinately to any straight line ma∣king an Angle with it, and representing the way of the Bodies mo∣tion, will designe a parallelogram which shall represent the Velo∣city of the whole Motion. But if the Impetus or Quickness of Moti∣on begin from Rest, and increase Uniformly, that is, in the same proportion continually with the times which are passed, the whole. Velocity of the Motion shall be represented by a Triangle, one side whereof is the whole time, and the other the greatest Impetus acquired in that time; or else by a parallelogram, one of whose sides is the whole time of Motion, and the other, half the greatest Impetus; or lastly by a parallelogram having for one side a mean proportional between the whole time & the half of that time, & for the other side the half of the greatest Impetus. For both these parallelograms are equal to one another, & severally equal to the triangle which is made of the whole line of time, and the greatest acquired Impetus; as is demonstrated in the Elements of Geometry.

2 In all Uniform Motions the Lengths which are transmit∣ted are to one another, as the product of the mean Impetus mul∣tiplyed into its time, to the product of the mean Impetus multiply∣ed also into its time.

For let AB (in the first Figure) be the Time, and AC the Impetus by which any Body passes with Uniform Motion through the Length DE; & in any part of the time AB, as in the time AF, let another Body be moved with Uniform Motion, first, with the same Impetus AC. This Body therefore in the time AB with the Impetus AC will pass through the length AF. Seeing therefore, when Bodies are moved in the same Time, & with the same Ve∣locity & Impetus in every part of their motion, the proportion of one Length transmitted to another Length trāsmitted, is the same w^th that of Time to Time, it followeth, that the Length transmitted in the time AB with the Impetus AC will be to the Length transmit∣ted in the time AF with the same Impetus AC, as AB it self is to AF, that is, as the parallelogram AI is to the parallelogram AH, that is, as the product of the time AB into the mean Impetus AC is to the product of the time AF into the same Impetus AC. Again, let it be supposed that a Body be moved in the time AF,

not with the same but with some other Uniform Impetus, as A L. Seeing therfore one of the Bodies has in all the parts of its motion the Impetus A C, and the other in like manner the Impetus A L, the Length trāsmitted by the Body moved with the Impetus A C will be to the Length transmitted by the Body moved with the Impetus A L, as A C it self is to A L, that is, as the parallelogram A H is to the parallelogram F L. Wherefore, by ordinate proportion it will be, as the parallelogram A I to the parallelogram F L, that is, as the product of the mean Impetus into the Time is to the product of the mean Impetus into the Time, so the Length transmitted in the time A B with the Impetus A C, to the length transmitted in the time A F with the Impetus, A L; which was to be demonstrated.

Corollary. Seeing therefore in Uniform Motion (as has been shewn) the Lengths transmitted are to one another as the paral∣lelograms which are made by the multiplication of the mean Im∣petus into the Times, that is, (by reason of the equality of the Im∣petus all the way) as the Times themselves, it will also be by per∣mutation, as to Time to Length, so Time to Length; and in gene∣ral, to this place are applicable all the properties and transmuta∣tions of Analogismes which I have set down and demonstrated in the 13 Chapter.

3 In Motion begun from Rest, and Uniformly Accelerated (that is, where the Impetus encreaseth continually according to the pro∣portion of the Times) it will also be, as one product made by the Mean Impetus multiplyed into the Time, to another product made likewise by the Mean Impetus multiplyed into the Time, so the Length transmitted in the one Time, to the Length transmitted in the other Time.

For let A B (in the same 1 figure) represent a Time; in the be∣ginning of which Time A, let the Impetus be as the point A; but as the Time goes on, so let the Impetus encrease Uniformly till in the last point of that Time A B, namely in B, the Impetus acquired be B I. Again, let A F represent another Time, in whose beginning A, let the Impetus be as the point it self A; but as the Time proceeds, so let the Impetus encrease Uniformly till in the last point F of the Time A F the Impetus acquired be F K; and let D E be the Length pas∣sed through in the Time A B with Impetus Uniformly encreased. I say the Length D E, is to the Length transmitted in the Time A F, as the Time A B multiplyed into the Mean of the Impetus encrea∣sing

through the time A B, is to the Time A F multiplyed into the Mean of the Impetus encreasing through the time A F.

For seeing the Triangle A B I is the whole Velocity of the Bo∣dy moved in the Time A B till the Impetus acquired be B I; and the Triangle A F K the whole Velocity of the Body moved in the Time A F with Impetus encreasing till there be acquired the Impe∣tus F K; the Length D E to the Length acquired in the Time A F with Impetus encreasing from Rest in A till there be acquired the Impetus F K, will be as the Triangle A B I to the Triangle A F K, that is, if the Triangles A B I and A F K be like, in duplicate proportion of the Time A B to the Time A F; but if un∣like, in the proportion compounded of the proportions of A B to B I, & of A K to A F. Wherefore, as A B I is to A F K, so let D E be to D P; for so, the Length transmitted in the Time A B with Impe∣tus encreasing to B I, will be to the Length transmitted in the Time A F with Impetus encreasing to F K, as the triangle A B I is to the triangle A F K; But the triangle A B I is made by the multiplicati∣on of the Time A B into the Mean of the Impetus encreasing to B I, and the triangle A F K is made by the multiplication of the Time A F into the Mean of the Impetus encreasing to F K; and therefore the Length D E which is transmitted in the Time A B with Impe∣tus encreasing to B I, to the Length D P which is transmitted in the Time A F with Impetus encreasing to F K, is as the product which is made of the Time A B multiplyed into its mean Impetus, to the product of the Time A F multiplyed also into its mean Im∣petus; which was to be proved.

• Corol. 1 In Motion Uniformly accelerated, the proportion of the Lengths transmitted, to that of their Times, is compounded of the proportions of their Times to their Times and Impetus to Impetus.
• Corol. 2 In Motion Uniformly accelerated, the Lengths trans∣mitted in equal times taken in continual succession from the be∣ginning of Motion, are as the differences of square numbers be∣ginning from Unity, namely, as 3, 5, 7, &c. For if in the first time the Length transmitted be as 1, in the first and second times the Length transmitted will be as 4, which is the Square of 2, and in the three first times, it will be as 9, which is the Square of 3, and in the four first times as 16, and so on. Now the differences of these Squares are 3, 5, 7, &c.
• Corol. 3 In Motion Uniformly accelerated from Rest, the Length

• transmitted, is to another Length transmitted vniformly in the same Time, but with such Impetus as was acquired by the acce∣lerated Motion in the last point of that Time, as a triangle to a pa∣rallelogram which have their altitude and base common. For see∣ing the Length D E (in the same 1 figure) is passed through with Velocity as the triangle A B I, it is necessary that for the passing through of a Length which is double to D E, the Velocity be as the parallelogram A I; for the parallelogram A I is double to the tri∣angle A B I.

4 In Motion which beginning from Rest, is so accelerated, that the Impetus thereof encrease continually in proportion duplicate to the proportion of the times in which it is made, a Length transmit∣ted in one time will be to a Length transmitted in another time, as the product made by the Mean Impetus multiplyed into the time of one of those Motions, to the product of the Mean Impetus multi∣plyed into the time of the other Motion.

For let A B (in the 2d. figure) represent a Time, in whose first in∣stant A let the Impetus be as the point A; but as the time proceeds, so let the Impetus encrease continually in duplicate proportion to that of the times, till in the last point of time B the Impetus acqui∣red be B I; then taking the point F any where in the time A B, let the Impetus F K acquired in the time A F be ordinately applyed to that point F. Seeing therefore the proportion of F K to B I is supposed to be duplicate to that of A F to A B, the proportion of A F to A B will be subduplicate to that of F K to B I; and that of A B to A F will be (by Chap. 13. Article 16) duplicate to that of B I to F K, and consequently the point K will be in a parabolical line whose diameter is A B and base B I; and for the same reason, to what point soever of the time A B the Impetus acquired in that time be ordinately applyed, the straight line designing that Impetus will be in the same parabolical line A K I. Wherefore the mean Impetus multiplyed into the whole time A B will be the Parabola A K I B, equal to the parallelogram A M, which parallelogram has for one side the line of time A B and for the other the line of the Impetus A L, which is two thirds of the Impetus B I; for every Parabola is equal to two thirds of that parallelogram with which it has its al∣titude and base common. Wherefore the whole Velocity in A B will be the parallelogram A M, as being made by the multiplicati∣on of the Impetus A L into the time A B. And in like manner, if F N

be taken, which is two thirds of the Impetus F K, and the parallelogram F O be completed, F O will be the whole Ve∣locity in the time A F, as being made by the Uniform Impe∣tus A O or F N multiplyed into the time A F. Let now the length transmitted in the time A B and with the Velocity A M be the straight line D E; and lastly, let the Length trans∣mitted in the time A F with the Velocity A N, be D P; I say that as A M is to A N, or as the Parabola A K I B to the Parabola A F K, so is D E to D P. For as A M is to F L (that is, as A B is to A F) so let D E be to D G. Now the proportion of A M to A N is com∣pounded of the proportions of A M to F L, and of F L to A N. But as A M to F L, so (by construction) is D E to D G; and as F L is to A N (seeing the time in both is the same, namely, A F), so is the Length D G to the Length D P; for Lengths transmitted in the same time are to one another as their Velocities are. Where∣fore by ordinate proportion, as A M is to A N, that is, as the mean Impetus A L multiplyed into its time A B, is to the mean Impetus A O multiplyed into A F, so is D E to D P; which was to be pro∣ved.

• Corol. 1 Lengths transmitted with Motion so accelerated that the Impetus encrease continually in duplicate proportion to that of their times, if the base represent the Impetus, are in triplicate pro∣portion of their Impetus acquired in the last point of their times. For as the Length D E is to the Length D P, so is the parallelo∣gram A M to the parallelogram A N, and so the Parabola A B I K to the Parabola A F K; But the proportion of the Parabola A B I K to the Parabola A F K is triplicate to the proportion which the base B I has to the base F K. Wherefore also the proportion of D E to D P, is triplicate to that of B I to F K.
• Corol. 2 Lengths transmitted in equal Times succeeding one another from the beginning, by Motion so accelerated, that the proportiō of the Impetus be duplicate to the proportiō of the times, are to one another as the differences of Cubique Numbers begin∣ning at Unity, that is, as 7, 19, 37, &c. For if in the first time the Length transmitted be as 1, the Length at the end of the second time will be as 8, at the end of the third time as 27, and at the end of the fourth time as 64, &c. which are Cubique Numbers, whose differences are 7, 19, 37, &c.
• Corol. 3 In Motion so accelerated, as that the Length transmitted

• be alwayes to the Length transmitted in duplicate proportion to their Times, the Length Uniformly transmitted in the whole time and with Impetus all the way equal to that which is last ac∣quired, is as a Parabola to a parallelogram of the same altitude & base, that is, as 2 to 3. For the Parabola A B I K is the Impetus en∣creasing in the time A B; and the parallelogram A I is the great∣est Uniform Impetus multiplyed into the same time A B. Where∣fore the Lengths transmitted will be as a Parabola to a parallelo∣gram &c. that is, as 2 to 3.

5 If I should proceed to the explication of such Motions as are made by Impetus encreasing in proportion triplicate, quadrupli∣cate, quintuplicate, &c. to that of their times, it would be a labour infinite and unnecessary. For by the same method by which I have computed such Lengths as are transmitted with Impetus encrea∣sing in single and duplicate proportion, any man may compute such as are transmitted with Impetus encreasing in triplicate, qua∣druplicate or what other proportion he pleases.

In making which computation he shall finde, that where the Impetus encrease in proportion triplicate to that of the times, there the whole Velocity will be designed by the first Parabolaster (of which see the next Chapter); and the Lengths transmitted will be in proportion quadruplicate to that of the times. And in like man∣ner, where the Impetus encrease in quadruplicate proportion to that of the times, that there the whole Velocity will be designed by the second Parabolaster, and the Lengths transmitted will be in quintuplicate proportion to that of the times; and so on conti∣nually.

6 If two Bodies with Uniform Motion transmit two Lengths, each with its own Impetus and Time, the proportion of the Lengths transmitted will be compounded of the proportions of Time to Time, and Impetus to Impetus, directly taken.

Let two Bodies be moved Uniformly (as in the 3d figure) One in the time A B with the Impetus A C, the other in the time A D with the Impetus A E. I say the Lengths transmitted have their proportion to one another compounded of the proportions of A B to A D, and of A C to A E. For let any Length whatsoever, as Z, be transmitted by one of the Bodies in the time A B with the Im∣petus A C; and any other Length, as X, be transmitted by the other Body in the time A D with the Impetus A E; and let the parallelo∣grams

A F and A G be completed. Seeing now Z is to X (by the 2d Article) as the Impetus A C multiplyed into the time A B is to the Impetus A E multiplyed into the time A D, that is, as A F to A G; the proportion of Z to X will be compounded of the same propor∣tions, of which the proportion of A F to A G is compounded; But the proportion of A F to A G is compounded of the proportions of the side A B to the side A D, and of the side A C to the side A E (as is evident by the Elements of Euclide), that is, of the pro∣portions of the time A B to the time A D, and of the Impetus A C to the Impetus A E. Wherefore also the proportion of Z to X is compounded of the same proportions of the time A B to the time A D, and of the Impetus A C to the Impetus A E; which was to be demonstrated.

• Corol. 1 When two Bodies are moved with Uniform Motion, if the Times and Impetus be in reciprocal proportion, the Lengths transmitted shall be equal. For if it were as A B to A D (in the same 3d figure) so reciprocally A E to A C, the proportion of A F to A G would be compounded of the proportions of A B to A D and of A C to A E, that is, of the proportions of A B to A D and of A D to A B. Wherefore, A F would be to A G as A B to A B, that is equal; and so the two products made by the multiplication of Impetus into Time would be equal; and by consequent, Z would be equal to X.
• Corol. 2 If two Bodies be moved in the same Time, but with different Impetus, the Lengths transmitted will be as Impetus to Im∣petus. For if the Time of both of them be A D, and their different Impetus be A E and A C, the proportion of A G to D C will be compounded of the proportions of A E to A C and of A D to A D, that is, of the proportions of A E to A C and of A C to A C; and so the proportion of A G to D C, that is, the proportion of Length to Length will be as A E to A C, that is, as that of Impetus to Impetus. In like manner, if two Bodies be moved Uniformly, and both of them with the same Impetus, but in different times, the proportion of the Lengths transmitted by them will be as that of their times. For if they have both the same Impetus A C, and their different times be A B & A D, the proportion ••f A F to D C will be compounded of the proportions of A B to A D and of A C to

• A C; that is, of the proportions of A B to A D and of A D, to A D; and therefore the proportion of A F to D C, that is, of Length to Length, will be the same with that of A B to A D, which is the proportion of Time to Time.

7 If two Bodies pass through two Lengths with Uniform Mo∣tion, the proportion of the Times in which they are moved will be compounded of the proportions of Length to Length and Impetus to Impetus reciprocally taken.

For let any two Lengths be given, as (in the same 3d figure) Z and X, and let one of them be transmitted with the Impetus A C, the other with the Impetus A E. I say the proportion of the Times in which they are transmitted, will be compounded of the propor∣tions of Z to X, and of A E (which is the Impetus with which X is transmitted) to A C (the Impetus with which Z is transmitted.) For seeing A F is the product of the Impetus A C multiplyed into the Time A B, the time of Motion through Z will be a line w^ch is made by the applicatiō of the parallelogram A F to the straight line A C, which line is A B; and therefore A B is the time of motion through Z. In like manner, seeing A G is the product of the Impetus A E multi∣plied into the Time A D, the time of motion through X wil be a line which is made by the application of A G to the straight line A D; but A D is the time of motiō through X. Now the proportion of A B to A D is cōpounded of the proportions of the parallelogram A F to the parallelogram A G, and of the Impetus A E to the Impetus A C; which may be demonstrated thus. Put the parallelograms in order A F, A G, D C; and it will be manifest that the propor∣tion of A F to D C is compounded of the proportions of A F to A G and of A G to D C; but A F is to D C as A B to A D; where∣fore also the proportion of A B to A D is compounded of the pro∣potrions of A F to A G & of A G to D C. And because the Length Z is to the Length X as A F is to A G, & the Impetus A E to the Impetus A C as A G to D C, therefore the proportion of A B to A D will be compounded of the proportions of the Length Z to the Length X, and of the Impetus A E to the Impetus A C; which was to be de∣monstrated.

In the same manner it may be proved, that in two Uniform Mo∣tions the proportion of the Impetus is compounded of the proporti∣ons,

of Length to Length, and of Time to Time reciprocally ta∣ken.

For if we suppose A C (in the same 3d figure) to be the Time, and A B the Impetus with which the Length Z is passed through; and A E to be the Time, and A D the Impetus with which the Length X is passed through, the demonstration will proceed as in the last Article.

8 If a Body be carried by two Movents together which move with straight and Uniform Motion, and concurre in any given angle, the line by which that Body passes will be a straight line.

Let the Movent A B (in the 4th figure) have straight and Uni∣form Motion, and be moved till it come into the place C D; and let another Movent A C, having likewise straight and Uniform Motion, and making with the Movent A B any given angle C A B, be understood to be moved in the same time to D B; and let the Body be placed in the point of their concourse A. I say the line which that Body describes with its Motion is a straight line. For let the parallelogram A B D C be completed, and its diagonal A D be drawn; and in the straight line A B let any point E be taken; and from it let E F be drawn parallel to the straight lines A C and B D, cutting A D in G; and through the point G let H I be drawn paral∣lel to the straight lines A B and C D; and lastly, let the measure of the time be A C. Seeing therefore both the Motions are made in the same time, when A B is in C D, the Body also will be in C D; and in like manner, when A C is in B D, the Body will be in B D. But A B is in C D at the same time when A C is in B D; and therefore the Body will be in C D and B D at the same time; Wherefore it will be in the common point D. Again, seeing the Motion from A C to B D is Uniform, that is, the Spaces transmit∣ted by it are in proportion to one another as the Times in which they are transmitted, when A C is in E F, the proportion of A B to A E will be same with that of E F to E G, that is, of the Time A C to the Time A H. Wherefore A B will be in H I in the same time in which A C is in E F, so that the Body will at the same time be in E F and in H I, and therefore in their common point G. And in the same manner it will, be wheresoever the point E be taken between A and B. Wherefore the Body will alwayes be in the Diagonal A D; which was to be demonstrated.

Corollary. From hence it is manifest, that the Body will be car∣ried through the same straight line A D, though the Motion be not Uniform, provided it have like acceleration; for the proportion of A B to A E will alwayes be the same with that of A C to A H.

9 If a Body be carried by two Movents together, which meet in any given angle, and are moved, the one Uniformly, the other with Motion Uniformly accelerated from Rest (that is, that the pro∣portion of their Impetus be as that of their Times) that is, that the proportion of their Lēgths be duplicate to that of the lines of their Times, till the line of greatest Impetus acquired by acceleration be equal to that of the line of Time of the Uniform Motion; the line in which the Body is carried will be the crooked line of a Semipa∣rabola, whose base is the Impetus last acquired, and Vertex the point of Rest.

Let the straight line A B (in the 5th Figure) be understood to be moved with Uniform Motion to C D; and let another Mo∣vent in the straight line A C be supposed to be moved in the same time to B D, but with motion Uniformly accelerted, that is, with such motion, that the proportion of the spaces which are trans∣mitted be alwayes duplicate to that of the Times, till the Impetus acquired be B D equal to the straight line A C; and let the Semi∣parabola A G D B be described. I say that by the concourse of those two Movents, the Body will be carried through the Semipa∣bolical crooked line A G D. For let the parallelelogram A B D C be completed; & from the point E taken any where in the straight line A B let E F be drawn parallel to A C, and cutting the crook∣ed line in G, and lastly, through the point G let A I be drawn pa∣rallel to the straight lines A B and C D. Seeing therefore the proportion of A B to A E is by supposition duplicate to the propor∣tion of E F to E G, that is, of the time A C to the time A H, at the same time when A C is in E F, A B will be in H I; and therefore the moved Body will be in the common point G. And so it will al∣wayes be in what part soever of A B the point E be taken. Where∣fore the moved Body will always be found in the parabolical line A G D; which was to be demonstr••ted.

10 If a Body be carried by two Movents together, which meet in any given angle, and are moved, the one Uniformly, the other with Impetus encreasing from Rest till it be equal to that of the U∣niform

Motion, and with such acceleration, that the proportion of the Lengths transmitted be every where triplicate to that of the Times in which they are transmitted, The line in which that Body is moved, will be the crooked line of the first Semiparabo∣laster of two Means, whose ba••e is the Impetus last acquired.

Let the straight line A B (in the 6th. Figure) be moved Uni∣formly to C D; and let another Movent A C be moved at the same time to B D with motion so accelerated, that the proportion of the Lengths transmitted by every where triplicate to the propor∣tion of their Times; and let the Impetus acquired in the end of that motion be B D, equal to the straight line A C; & lastly, let A D be the crooked line of the first Semiparabolaster of two Means. I say that by the concourse of the two Movents together, the Body will be alwayes in that crooked line A D. For let the parallelogram A B D C be completed; and from the point E taken any where in the straight line A B let E F be drawn parallel to A C and cutting the crooked line in G; and through the point G let H I be drawn parallel to the straight lines A B and C D. Seeing therefore the proportion of A B to A E is (by supposition) triplicate to the pro∣portion of E F to E G, that is, of the time A C to the time A H, at the same time when A C is in E F, A B will be in H I; and there∣fore the moved Body will be in the common point G. And so it will alwayes be in what part soever of A B the point E be taken; and by consequent the Body will always be in the crooked line A G D; which was to be demonstrated.

11 By the same method it may be shewn what line it is that it made by the motion of a Body carried by the concourse of any two Movents, which are moved, one of them Uniformly, the o∣ther with acceleration, but in such proportions of Spaces and Times as are explicable by Numbers, as duplicate, triplicate &c. or such as may be designed by any broken number whatsoever. For which this is the Rule.

Let the two numbers of the Length & Time be added together; & let their Sum be the Denominator of a Fraction, whose Nume∣rator must be the number of the Length. Seek this Fraction in the Table of the third Article of the 17th Chapter, and the line sought will be that which denominates the three-sided Figure no∣ted on the left hand, and the kind of it will be that which is num∣bred

above over the Fraction. For example, Let there be a con∣course of two Movements, whereof one is moved Uniformly, the other with motion so accelerated that the Spaces are to the Times as 5 to 3. Let a Fraction be made whose Denominator is the Sum of 5 and 3, and the Numerator 5, namely the Fraction ⅝. Seek in the Table, and you will find ⅝ to be the third in that row which be∣longs to the three-sided Figure of four Means. Wherfore the line of Motion made by the concourse of two such Movents as are last of all described, will be the crooked line of the third Parabolaster of four Means.

12 If Motion be made by the concourse of two Movents where∣of one is moved Uniformly, the other beginning from Rest in the Angle of concourse with any acceleration whatsoever; the Movent which is Moved Uniformly shall put forward the moved Body in the several parallel Spaces, lesse, then if both the Movents had U∣niform motion; and still lesse and lesse, as the Motion of the other Movent is more and more accelerated.

Let the Body be placed in A (in the 7th figure) and be mo∣ved by two Movents, by one with Uniform Motion from the straight line A B to the straight line C D parallel to it; and by the other with any acceleration from the straight line A C to the straight line B D parallel to it; and in the parallelogram A B D C let a Space be taken between any two parallels E F and G H. I say, that whilest the Movent A C passes through the latitude which is between E F and G H, the Body is lesse moved forwards from A B towards C D, then it would have been, if the Motion from A C to B D had been Uniform.

For suppose that whilest the Body is made to descend to the parallel E F by the power of the Movent from A C towards B D, the same Body in the same time is moved forwards to any point F in the line E F by the power of the Movent from A B towards C D; and let the straight line A F be drawn and produced inde∣terminately, cutting G H in H. Seeing therefore it is as A E to A G, so E F to G H; if A C should descend towards B D with uniform Motion, the Body in the time G H (for I make A C and its pa∣rallels the measure of time) would be found in the point H. But be∣cause A C is supposed to be moued towards B D which motion continually accelerated, that is, in greater proportion of Space to

Space then of Time to Time, in the time G H the Body will be in some parallel beyond it, as between G H and B D. Suppose now that in the end of the time G H it be in the parallel I K, & in I K let I L be taken equal to G H. When therefore the Body is in the pa∣rallel I K, it will be in the point L. Wherefore when it was in the parallel G H, it was in some point between G and H, as in the point M; but if both the Motions had been Uniform it had been in the point H; and therefore whilest the Movent A C passes over the latitude which is between E F and G A, the Body is less moved forwards from A B towards C D, then it would have been if both the Motions had been Uniform; which was to be demonstrated.

13 Any Length being given whch is passed through in a given time with uniform motion, To find out what Length shall be pas∣sed through in the same time with Motion uniformly accelerated, that is, with such Motion, that the proportion of the Lengths pas∣sed through be continually duplicate to that of their Times, and that the line of the Impetus last acquired, be equal to the line of the whole time of the Motion.

Let A B (in the 8th. figure) be a Length transmitted with Uni∣form Motion in the time A C; and let it be required to find ano∣ther Length which shall be transmitted in the same time with Motion Uniformly accelerated, so, that the line of the Impetus last acquired be equal to the straight line A C.

Let the parallelogram A B D C be completed; and let B D be divided in the middle at E; and between B E and B D let B F be a mean proportional; and let A F be drawn and produced till it meet with C D produced in G; and lastly, let the parallelogram A C G H be completed. I say A H is the Length required.

For as duplicate proportion is to single proportion, so let A H be to A I, that is, let A I be the half of A H; and let I K be drawn parallel to the straight line A C, and cutting the Diagonal A D in K, and the straight line A G in L. Seeing therefore A I is the half of A H, I L will also be the half of B D, that is, equal to B E, and I K equal to B F; for B D, (that is, G H), B F, and B E (that is, I L) being continual proportionals, A H, A B, and A I will also be continual proportionals. But as A B is to A I, that is, as A H is to A B, so is B D to I K, and so also is G H, that is, B D to B F; and therefore B F and I K are equal. Now the proportion of A H to A I is duplicate to the proportion of A B to A I, that is, to that

of BD to IK, or of GH to IK. Wherefore the point K will be in a Parabola, whose diameter is AH & ba••e GH, which GH is equal to AC. The Body therefore proceeding from Rest in A with motion Uniformly accelerated in the time AC, when it has passed through the Length AH, will acquire the Impetus GH equal to the time AC; that is, such Impetus, as that with it the Body will pass through the Length AC in the time AC. Wherefore any Length being given, &c. which was propounded to be done.

14 Any Length being given which in a given Time is transmitted with Uniform Motion, To find out what Length shall be transmit∣ted in the same Time with Motion so accelerated, that the Lengths transmitted be continually in triplicate proportion to that of their Times, and the line of the Impetus last of all acquired be equal to the Line of Time given.

Let the given Length AB (in the 9th figure) be transmitted with Uniform motion in the Time AC; and let it be required to find what Length shall be transmitted in the same time with mo∣tion so accelerated, that the Lengths transmitted be continually in triplicate proportion to that of their Times, and the Impetus last acquired be equal to the Time given.

Let the parallelogram ABDC be completed; and let BD be so divided in E, that BE be a third part of the whole BD; and let BF be a mean proportional between BD and BE; and let AF be drawn and produced till it meet the straight line CD in G; and lastly, let the parallelogram ACGH be completed. I say AH is the Length required.

For as triplicate proportion is to single proportion, so let AH be to another line AI, that is, make AI a third part of the whole AH; and let IK be drawn parallel to the straight line AC, cutting the Diagonal AD in K, and the straight line AG in L; then, as AB is to AI, so let AI be to another AN; and from the point N let NQ be drawn parallel to AC, cutting AG, AD, and FK produ∣ced, in P, M and O; and last of all let FO and LM be drawn, which will be equal and parallel to the straight lines BN and IN. By this construction, the Lengths transmitted AH, AB, AI and AN will be continual proportionals; and in like manner, the Times GH, BF, IL and NP, that is, NQ, NO, NM and NP will be continual proportionals, and in the same proportion with

AH, AB, AI, and AN. Wherefore the proportion of AH to AN is the same with that of BD, that is, of NQ to NP; and the proportion of NQ to NP triplicate to that of NQ to NO, that is, triplicate to that of BD to IK; Wherefore also the Length AH is to the Length AN in triplicate proportion to that of the Time BD to the Time IK; and therefore the crooked line of the first three sided figure of two means, whose Diameter is AH, and base GH equal to AC, shall pass through the point O; and con∣sequently AH shall be transmitted in the time AC, and shall have its last acquired Impetus GH equal to AC, and the propor∣tions of the Lengths acquired in any of the times triplicate to the proportions of the times themselves. Wherefore AH is the Length required to be found out,

By the same method, if a Length be given which is transmit∣ted with Uniform Motion in any given Time, another Length may be found out, which shall be transmitted in the same Time with motion so accelerated, that the Lengths transmitted shall be to the Times in which they are transmitted, in proportion quadrupli∣cate, quintuplicate, and so on infinitely. For if BD be divided in E, so, that BD be to BE as 4 to 1; and there be taken between BD and BE a mean proportional FB; and as AH is to AB, so AB be made to a third, and again so that third to a fourth, and that fourth to a fifth AN, so that the proportion of AH to AN be qua∣druplicate to that of AH to AB, and the parallelogram NBFO be completed; the crooked line of the first three-sided Figure of three Means will pass through the point O▪ and consequently the Body moved will acquire the Impetus GH equal to AC in the time AC. And so of the rest.

15 Also, if the proportion of the Lengths transmitted, be to that of their Times, as any number to any number, the same me∣thod serves for the finding out of the Length transmitted with such Impetus, and in such Time.

For let AC (in the 10 figure) be the time, in which a Body is transmitted with Uniform Motion from A to B; and the paralle∣logram ABDC being completed, let it be required to find out a Length in which that Body may be moved in the same time AC, frō A w^th motion so accelerated, that the proportion of the Lengths transmitted, to that of the Times be continually as 3 to 2.

Let BD be so divided in E, that BD be to BE as 3 to 2; and be∣tween BD and BE let BF be a mean proportionall; and let AF be drawn and produced till it meet with CD produced in G; and making AM a mean proportional between AH and AB, let it be as AM to AB, so AB to AI; and so the proportion of AH to AI will be to that of AH to AB, as 3 to 2. (for of the proportions of which that of AH to AM is one, that of AH to AB is two, and that of AH to AI is three;) & consequently as 3 to 2 to that of GH to BF, & (FK being drawn parallel to BI, and cutting AD in K) so likewise to that of GH or BD to IK; Wherefore the propor∣tion of the Length AH to AI is to the proportion of the Time BD to IK, as 3 to 2; and therefore, if in the time AC, the Body be moved with accelerated motion, as was propounded, till it ac∣quire the Impetus HG equal to AC, the Length transmitted in the same Time will be AH.

16 But if the proportion of the Lengths to that of the Times had been as 4 to 3, there should then have been taken two mean proportionals between AH and AB, and their proportion should have been continued one term further, so that AH to AB might have three of the same proportions, of which AH to AI has four; and all things else should have been done as is already shewn. Now the way how to interpose any number of Means between two Lines given, is not yet found out. Nevertheless, this may stand for a general Rule; If there be a Time given, and a Length be transmitted in that Time with Uniform Motion; as for example, if the Time be AC, and the Length AB; the straight line AG, which determines the Length CG or AH transmitted in the same Time AC with any accelerated motion, shall so cut BD in F, that BF shall be a mean proportional between BD and BE, BE being so taken in BD, that the proportion of Length to Length be every where to the proportion of Time to Time, as the whole BD is to its part BE.

17 If in a given Time, two Lengths be transmitted, One with uniform motion, the other with motion accelerated in any propor∣tion of the Lengths to the Times; and again in part of the same Time, parts of the same Lengths be transmitted with the same motions, the whole Length will exceed the other Length in the same proportion in which one part exceeds the other part.

For example, let AB (in the 8th. figure) be a Length transmit∣ted

in the time AC with uniform Motion; and let AH be another Length transmitted in the same time with Motion uniformly ac∣celerated, so that the Impetus last acquired be GH equal to AC; and in AH let any part AI be taken, and transmitted in part of the time AC with uniform Motion; and let another part AB be taken, and transmitted in the same part of the time AC with Motion u∣niformly accelerated. I say, that as AH is to AB, so will AB be to AI.

Let BD be drawn parallel and equal to HG, and divided in the midst at E, and between BD and BE, let a mean proportional be taken as BF; & the straight line AG (by the demonstration of the 13th Art.) shal pass through F. And dividing AH in the midst at I, AB shall be a mean proportional between AH and AI. Again (because AI and AB are described by the same Motions) if IK be drawn parallel and equal to BF or AM, and divided in the midst at N, and between IK and IN be taken the mean proportional IL, the straight line AF will (by the demonstration of the same 13th Ar••.) pass through L. And dividing AB in the midst at O, the line AI will be a mean proportional between AB and AO. Where▪ AB is divided in I and O in like manner as AH is divided in B and I; and as AH to AB so is AB to AI. Which was to be proved.

Coroll. Also as AH to AB, so is HB to BI; and so also BI to IO.

And as this (where one of the Motions is uniformly accelerated) is proved out of the demonstration of the 13th Article; so (when the accelerations are in double proportion to the times) the same may be proved by the demonstration of the 14th Art. and by the same method in all other accelerations, whose proportions to the times are explicable in numbers.

18 If two sides which contain an Angle in any Parallelogram, be moved in the same time to the sides opposite to them, one of them with Uniform Motion, the other with Motion Uniformly accelera∣ted; that side which is moved Uniformly will effect as much with its concourse through the whole Length transmitted, as it would do if the other Motion were also Uniform, and the Length trans∣mitted by it in the same time were a mean proportional between the whole and the half.

Let the side AB of the Parallelogram ABDC (in the 11th Figure) be understood to be moved with Uniform Motion till it be coincident with CD; and let the time of that Motion be AC

or BD. Also in the same time let the side AC be understood to be moved with Motion uniformly accelerated, till it be coincident with BD; then dividing AB in the middle in E, let AF be made a mean proportional between AB and AE; and drawing FG paral∣lel to AC, let the side AC be understood to be moved in the same time AC with uniform Motion till it be coincident with FG. I say the whole AB confers as much to the velocity of the Body placed in A when the Motion of AC is uniformly accelerated till it come to BD, as the part AF confers to the same when the side AC is mo∣ved Uniformly and in the same time to FG.

For seeing AF is a mean proportional between the whole AB & it is half AE, BD wil (by the 13th Art.) be the last Impetus acquired by AC with motion uniformly accelerated till it come to the same BD; and consequently the straight line FB will be the excess by w^ch the Length transmitted by AC with motion uniformly accele∣rated, will exceed the Length transmitted by the same AC in the same time with Uniform Motion, and with Impetus every where e∣qual to BD. Wherefore if the whole AB be moved Uniformly to CD in the same time in which AC is moved Uniformly to FG, the part FB (seeing it concurs not at all with the Motion of the side AC which is supposed to be moved onely to FG) will cō∣fer nothing to its motion. Again, supposing the side AC to be mo∣ved to BD with Motion Uniformly accelerated, the side AB with its uniform Motion to CD will less put forwards the Body when it▪ is accelerated in all the parallels, then when it is not at all accelera∣ted; & by how much the greater the acceleration is, by so much the less it will put it forwards (as is shewn in the 12th Artic.) When therefore AC is in FG with accelerated Motion, the Body will not be in the side CD at the point G, but at the point D; so that GD wil be the excess by which the Length transmitted with accelera∣ted Motion to BD, exceeds the Length transmitted with Uniform Motion to FG; so that the Body by its acceleration avoids the acti∣on of the part AF, & comes to the side CD in the time AC, and makes the Length CD, which is equal to the Length AB. Where∣fore Uniform Motion from AB to CD in the time AC works no more in the whole Length AB upon the Body uniformly accelera∣ted from AC to BD, then if AC were moved in the same time with uniform Motion to FG; the difference consisting onely in this, that when AB works upon the Body uniformly moved from AC to

FG, that by which the accelerated Motion exceeds the Uniform Motion, is altogether in FB, or GD; but when the same AB works upon the Body acclerated, that by which the accelerated Motion exceeds the Uniform Motion, is dispersed through the whole Length AB or CD, yet so that if it were collected and put together, it would be equal to the same FB or GD. Wherefore, If two sides which contain an angle &c; which was to be demon∣strated.

19 If two transmitted Lengths have to their Times any other proportion explicable by number, & the side AB be so divided in E, that AB be to AE in the same proportion which the Lengths transmitted have to the Times in which they are transmitted, and between AB and AE there be taken a mean proportional AF, it may be shewn by the same method, that the side which is moved with Uniform Motion, works as much with its concourse through the whole Length AB, as it would do if the other Motion were also Uniform, and the Length transmitted in the same Time AC were that mean proportional AF.

And thus much concerning Motion by concourse.

CHAP. XVII. Of Figures Deficient.

• 1 Definitions of a Deficient Figure; of a Complete Figure; of the Complement of a Deficient Figure; and of Pro••o••tions which are Proportional and Com∣mensurable to one another.
• 2 The proportion of a Deficient Figure to its Complement.
• 3 The proportions of Deficient Figures to the Parallelograms in which they are described, set forth in a Table.
• 4 The Description and Pro∣duction of the same Figures.
• 5 The drawing of Tangents to them.
• 6 In what proportion the same Figures exceed a straight lined Triangle of the same Altitude and Base.
• 7 A Table of Solid Deficient Figures described in a Cylinder.
• 8 In what proportion the same Figures exceed a Cone of the same Altitude and Base.
• 9 How a plain Deficient Figure may be described in a Parallelogram, so, that it be to a Triangle of the same Base and Al∣titude, as another Deficient Figure (plain or solid) twice taken, is to the same Deficient Figure together with the Complete Figure in which it is de∣scribed.
• 10 The transferring of certain properties of Deficient Figures described in a Parallelogram to the proportions of the Spaces transmitted with several degrees of Velocity.
• 11 Of Deficient Figures described in a Circle.
• 12 The proposition demonstrated in the 2d. Article, confirmed from the E∣lements of Philosophy.
• 13 An unusual way of reasoning concerning the E∣quality between the superficies of a portion of a Sphere, and a Circle.
• 14 How from the description of Deficient Figures in a Parallelogram, any number of mean Proportionals may be found out between two given straight lines.

1 I Call those Deficient Figures, which may be under∣stood to be generated by the Uniform Motion of some quantity, which decreases continually, till at last it have no magnitude at all.

And I call that a Complete Figure, answering to a Deficient Figure, which is generated with the same motion, and in the same time, by a Quantity which retaines alwayes its whole magnitude.

The Complement of a Deficient Figure is that, which being ad∣ded to the Deficient Figure, makes it Complete.

Four Proportions are said to be Proportionall, when the first of

them is to the second, as the third is to the fourth. For exam∣ple, if the first proportion be duplicate to the second; and a∣gain the third be duplicate to the fourth, those Proportions are said to be Proportionall.

And Commensurable Proportions are those, which are to one a∣nother as number to number. As when to a proportion given, one proportion is duplicate, another triplicate, the duplicate pro∣portion will be to the triplicate proportion as 2 to 3; but to the given proportion it will be as 2 to ••; and therefore I call those three proportions Commensurable.

2 A Deficient Figure, which is made by a Quantity continu∣ally decreasing to nothing by proportions every where proportio∣nall and commensurable, is to its Complement, as the propor∣tion of the whole altitude, to an altitude diminished in any time, is to the proportion of the whole Quantity which describes the Figure, to the same Quantity diminished in the same time.

Let the quantity AB (in the 1 figure) by its motion through the altitude AC, describe the Complete Figure AD; and againe, let the same quantity, by decreasing continually to nothing in C, describe the Deficient Figure ABEFC, whose Complement will be the Figure BDCFE. Now let AB be supposed to be moved till it lie in GK, so that the altitude diminished be GC, and AB diminished be GE; and let the proportion of the whole alti∣tude AC to the diminished altitude GC, be (for example) tripli∣cate to the proportion of the whole quantity AB or GK, to the di∣minished quantity GE. And in like manner, let HI be taken e∣qual to GE, & let it be diminished to HF; and let the proportion of GC to HC be triplicate to that of HI to HF; & let the same be done in as many parts of the straight line AC as is possible; and a line be drawn through the points B, E, F and C. I say the Deficient Figure ABEFC, is to its Complement BDCEF as 3 to ••, or as the proportion of AC to GC is to the proportion of AB, that is, of GK to GE.

For (by the second Article of the 15. Chap.) the proportion of the complement BEFCD to the deficient figure ABEFC, is all the proportions of DB to OE, and of DB to QF, and of all the lines parallel to DB terminated in the line BEFC, to all the parallels to AB terminated in the same points of the line BEFC.

And seeing the proportions of DB to OE, and of DB to QF &c are every where triplicate of the proportions of AB to GE, and of AB to HF &c. the proportions of HF to AB, and of GE to AB &c. (by the 16 Article of the 13 Chap.) are triplicate of the proportions of QF to DB, and of OE to DB &c. and therefore the deficient figure ABEFC which is the aggregate of all the lines HF, GE, AB, &c. is triple to the complement BEFCD made of all the lines QF, OE, DB, &c. which was to be proved.

It follows from hence, That the same complement BEFCD is ¼ of the whole Parallelogram. And by the same method may be calculated in all other Deficient Figures generated as above de∣clared, the proportion of the Parallelogram to either of its parts; as that when the parallels encrease fom a point in the same pro∣portion, the Parallelogram will be divided into two equal Trian∣gles; when one encrease is double to the other, it will be divided into a Semiparabola and its Complement, or into 2 and 1.

The same construction standing, the same conclusion may o∣therwise be demonstrated, thus.

Let the straight line CB be drawn cutting GK in L, & through L let MN be drawn parallel to the straight line AC; wherefore the Parallelograms GM and LD will be equal. Then let LK be divided into three equal parts, so that it may be to one of those parts in the same proportion which the proportion of AC to GC or of GK to GL hath to the proportion of GK to GE. Therefore LK will be to one of those three parts as the Arithmetical proportion between GK and GL is to the Arithmetical proportion between GK and the same GK want the third part of LK; and KE will be somwhat greater then a third of LK. Seeing now the altitude AG or ML is by reason of the continual decrease, to be supposed less then any quantity that can be given; LK (which is intercepted between the Diagonal BC and the side BD) will be also less then any quantity that can be given; and consequently, if G be put so neer to A in g, as that the difference between Cg and CA be less then any quantity that can be assigned, the diffe∣rence also between Cl (removing L to l) and CB, will be less then any quantity that can be assigned; and the line gl be∣ing drawn & produced to the line BD in k cutting the crooked line

in e, the proportion of Gk to Gl will still be triplicate to the pro∣portion of Gk to Ge, and the difference between k and e the third part of kl will be less then any quantity that can be given; and therefore the Parallelogram eD will differ from a third part of the Parallelogram Ae by a less difference then any quantity that can be assigned. Again, let HI be drawn parallel and equal to ge, cutting CB in P, the crooked line in F, and BD in I, and the proportion of Cg, to CH will be triplicate to the pro∣portion of HF to HP, and IF will be greater then the third part of PI. But again, setting H in h so neer to g, as that the difference between Ch and Cg may be but as a point, the point P will also in p be so neer to l, as that the difference between Cp and Cl will be but as a point; and drawing hp till it meet with gk in i, cutting the crooked line in f, and having drawn eo parallel to BD, cutting DC in o, the Parallelogram fo will differ less from the third part of the Parallelogram gf, then by any quantity that can be given. And so it will be in all other Spaces generated in the same man∣ner. Wherefore the differences of the Arithmetical and Geome∣trical Means, which are but as so many points B, e, f, &c. (seeing the whole Figure is made up of so many indivisible Spaces) will constitute a certain line, such as is the line BEFC, which will di∣vide the complete Figure AD into two parts; whereof one, name∣ly ABEFC, which I call a Deficient Figure, is triple to the o∣ther, namely BDCEF, which I call the Complement thereof. And whereas the proportion of the altitudes to one another, is in this case everywhere triplicate to that of the decreasing quantities to one another; in the same manner if the proportion of the al∣titudes had been every where quadruplicate to that of the de∣creasing quantities it might have been demonstrated, that the Deficient Figure had been quadruple to its Complement; and so in any other proportion, Wherefore, a Deficient Figure, which is made, &c. Which was to be demonstrated.

The same rule ••oldeth also in the diminution of the Bases of Cylinders, as is demonstrated Chap. 15. Art. 2.

•• By this Proposition, the magnitudes of all Deficient Figures (when the proportions by which their bases decrease continually, are proportionall to those by which their altitudes decrease) may be compared with the magnitudes of their Complements;

and consequently, with the magnitudes of their Complete Figures. And they will be found to be as I have set them down in the fol∣lowing Tables; in which I compare a Parallelogram with three∣sided Figures; and first with a straight lined triangle, made by the base of the Parallelogram continually decreasing in such manner; that the altitudes be alwayes in proportion to one another as the bases are, and so the triangle will be equal to its Complement; or the proportions of the altitudes and bases wil be as 1 to 1, and then the triangle will be half the Parallelogram. Secondly, with that three-sided Figure which is made by the continual decreasing of the bases in subduplicate proportion to that of the altitudes; and so the Deficient Figure will be double to its Complement, and to the Parallelogram as 2 to 3. Then, with that, where the proportion of the altitudes is triplicate to that of the bases; and then the De∣ficient Figure will be triple to its Complement, and to the Paral∣lelogram as 3 to 4. Also the proportion of the altitudes to that of the bases may be as 3 to 2; and then the Deficient Figure will be to its Complement as 3 to 2, & to the Parallelogram as 3 to 5; and so forwards according as more mean proportionals are taken, or as the proportions are more multiplyed, as may be seen in the follow∣ing Table. For example, if the bases decrease so, that the pro∣portion of the altitudes to that of the bases be alwayes as 5 to 2, and it be demanded what proportion the Figure made has to the Parallelogram, which is supposed to be Unity; then, seeing that where the proportion is taken five times, there must be four Means; look in the Table amongst the three-sided figures of four Means, and seeing the proportion was as 5 to 2, look in the upper∣most row for the number 2, and descending in the 2d Columne till you meet with that three-sided Figure, you will finde 5/7; which shews that the Deficient Figure is to the Parallelogram as 5/•• to 1, or as 5 to 7.

	1	2	3	4	5	6	7
Parallelogram: . . . . . . . . . .	1
Straight-sided Triangle . . . . .	½
Three-sided figure of 1 Mean	⅔
Three-sided figure of 2 Means	¾	⅗
Three-sided figure of 3 Means	⅘	4/6	4/7
Three-sided figure of 4 Means	⅚	5/7	⅝	5 9
Three-sided figure of 5 Means	6/7	6/8	6/9	6/10	6/11
Three-sided figure of 6 Means	⅞	7/9	7/10	7/11	7/12	7/13
Three-sided figure of 7 Means	8/9	8/10	8/11	8/12	8/13	8/14	8/15

4 Now for the better understanding of the nature of these three-sided figures, I will shew how they may be described by points; and first, those which are in the first column of the Table. Any Parallelogram being described, as ABCD (in the 2d. fi∣gure,) let the Diagonal BD be drawn; and the straight-lined tri∣angle BCD will be half the Parallelogram; Then let any number of lines, as EF, be drawn parallel to the Side BC, and cutting the Diagonal BD in G; & let it be every where, as EF to EG, so EG to another EH; and through all the points H let the line BHHD be drawn; and the Figure BHHDC will be that which I call a Three-sided Figure of one Mean, because in three proportionals, as EF, EG and EH, there is but one Mean, namely, EG; and this three-sided figure will be ⅔ of the Parallelogram, and is called a Parabola. Again, let it be as EG to EH, so EH to another EI, and let the line BIID be drawn, making the three-sided figure BIIDC; & this will be ¾ of the Parallelogram, and is by many called a Cubique Parabola. In like manner, if the proportions be fur∣ther continued in EF, there will be made the rest of the three∣sided

figures of the first Column; which I thus demonstrate. Let there be drawn straight lines, as HK and GL parallel to the base DC. Seeing therefore the proportion of EF to EH is duplicate of that of EF to EG, or of BC to BL, that is, of CD to LG, or of KM (producing KH to AD in M) to KH, the proportion of BC to BK will be duplicate to that of KM to KH; but as BC is to BK, so is DC, or KM to KN; and therefore the proporti∣on of KM to KN is duplicate to that of KM to KH; and so it will be wheresoever the parallel KM be placed. Wherefore the Figure BHHDC is double to its Complement BHHDA, and conse∣quently ⅔ of the whole Parallelogram. In the same manner if through I, be drawn OPIQ parallel and equal to CD, it may be demonstrated that the proportion of OQ to OP, that is, of BC to BO, is triplicate to that of OQ to OI, and therefore that the Fi∣gure BIIDC is triple to its Complement BIIDA, and conse∣quently ¾ of the whole Parallelogram, &c.

Secondly, such three-sided figures as are in any of the transverse rowes, may be thus described. Let ABCD (in the 3d. Figure) be a Parallelogram, whose Diagonal is BD. I would describe in it such figures, as in the preceding Table I call Three-sided Figures of three Means. Parallel to DC, I draw EF as often as is necessary, cutting BD in G; and between EF and EG I take three proportionals EH, EI and EK. If now there be drawn lines through all the points H, I & K; that through all the points H will make the figure BHDC, which is the first of those three-sided figures; and that through all the points I, will make the figure BIDC, which is the second; and that which is drawn through all the points K, will make the figure BKDC the third of those three-sided figures. The first of these (seeing the proportion of EF to EC is quadruplicate of that of EF to EH) will be to its Complement as 4 to 1, and to the Parallelogram as 4 to 5. The se∣cond (seeing the proportion of EF to EG is to that of EF to EI as 4 to 2) will be double to its Complement, and 4/6 or ⅔ of the Parallelogram. The third (seeing the proportion of EF to EG is to that of EF to EK as 4 to 3) will be to its Complement as 4 to 3, and to the Parallelogram as 4 to 7.

Any of these, figures being described, may be produced at pleasure, thus; Let ABCD (in the 4th figure) be a Parallelogram, and in it let the figure BKDC be described, namely, the third three-sided figure of three Means. Let BD be produced indefi∣nitely to E, and let EF be made parallel to the base DC, cutting

AD produced in G, and BC produced in F; and in GE let the point H be so taken, that the proportion of FE to FG may be quadruplicate to that of FE to FH (which may be done by ma∣king FH the greatest of three proportionals between FE and FG); the crooked line BKD produced, will pass through the point H. For if the straight line BH be drawn, cutting CD in I, and HL be drawn parallel to GD, and meeting CD produced in L; it will be as FE to FG, so CL to CI; that is, in quadruplicate proportion to that of FE to FH, or of CD to CI. Wherefore if the line BKD be produced according to its generation, it will fall upon the point H.

5 A straight line may be drawn so, as to touch the crooked line of the said figure in any point, in this manner. Let it be required to draw a Tangent to the line BKDH (in the 4th figure) in the point D. Let the points B and D be connected, and drawing DA equal and parallel to BC, let B and A be connected; and be∣cause this figure is by construction the third of three Means, let there be taken in AB three points, so, that by them the same AB be divided into four equal parts; of which take three, namely, AM, so that AB may be to AM, as the figure BKDC is to its Complement. I say the straight line MD, will touch the figure in the point given D. For let there be drawn any where between AB and DC a parallel, as RQ, cutting the straight line BD, the crooked line BD, the straight line MD, and the straight line AD in the points P, K, O and Q. RK will therefore (by con∣struction) be the least of three Means in Geometrical propor∣tion between RQ and RP. Wherefore (by the Coroll. of the 28th Article of the 13th Chapter) RK will be less then RO; and there∣fore MD will fall without the figure. Now if MD be produced to N, FN will be the least of three Means in Arithmetical pro∣portion between FE and FG; and FH will be the greatest of three Means in Geometrical proportion between the same FE and FG. Wherefore (by the same Coroll. of the 28 Artic. of the 13th Chap.) FH will be less then FN; and therefore DN will fall without the figure, and the straight line MN will touch the same figure one∣ly in the point D.

6 The proportion of a Deficient Figure to its Complement being known, it may also be known what proportion a straight∣lined

triangle has to the excess of the Deficient Figure above the same triangle; and these proportions I have set down in the follow∣ing Table; where if you seek (for example) how much the fourth three-sided figure of five Means exceeds a triangle of the same al∣titude and base, you will find in the concourse of the fourth column with the three-sided figures of five Means, 2/10; by which is signi∣fied, that that three-sided figure exceeds the triangle by two tenths, or by one fifth part of the same triangle.

	1	2	3	4	5	6	7
The Triangle . . . . . . . . . . . .	1
The Excess of a Three-sided figure of one Mean	⅓
The Excess of a Three-sided figure of 2 Means	2/4	⅕
The Excess of a Three-sided figure of 3 Means	⅗	2/6	1/7
The Excess of a Three-sided figure of 4 Means	4/6	3/7	2/8	1/9
The Excess of a Three-sided figure of 5 Means	5/7	4/8	3/9	2/10	1/11
The Excess of a Three-sided figure of 6 Means	6/8	5/9	4/10	3/11	2/12	1/13
The Excess of a Three-sided figure of 7 Means	7/9	6/10	5/11	4/12	3/13	2/14	1/15

7 In the next Table are set down the proportion of a Cone, and the Solids of the said three-sided figures, namely, the proportions between them and a Cylinder. As for example, in the concourse of the second Column with the three-sided figures of four Means, you have ••/9; which gives you to understand, that the Solid of the second three-sided figure of four Means is to the Cylinder as ••/9 to 1, or as 5 to 9,

	1	2	3	4	5	6	7
The Cylinder	1
A Cone	⅓
The Solids of a Three-sided figure of one Mean	2/4
The Solids of a Three-sided figure of 2 Means	⅗	3/7
The Solids of a Three-sided figure of 3 Means	4/6	4/8	4/10
The Solids of a Three-sided figure of 4 Means	5/7	5/9	5/11	5/13
The Solids of a Three-sided figure of 5 Means	6/8	6/10	6/12	6/14	6/16
The Solids of a Three-sided figure of 6 Means	7/9	7/11	7/13	7/15	7/17	7/19
The Solids of a Three-sided figure of 7 Means	8/10	8/12	8/14	8/16	8/18	8/20	8/22

8 Lastly, the Excess of the Solids of the said three-sided fi∣gures, above a Cone of the same altitude and base, are set down in the Table which follows

	1	2	3	4	5	6	7
The Cone	1
The Exces∣ses of the Solids of these three∣sided figures above a C n•• Of the Solid of a three-si∣ded figure of one Mean	6/12
The Exces∣ses of the Solids of these three∣sided figures above a C n•• Of the Solid of a three-si∣ded figure of 2 Means	12/15	6/21
The Exces∣ses of the Solids of these three∣sided figures above a C n•• Of the Solid of a three-si∣ded figure of 3 Means	18/18	12/24	6/30
The Exces∣ses of the Solids of these three∣sided figures above a C n•• Of the Solid of a three-si∣ded figure of 4 Means	24/21	18/27	12/33	6/39
The Exces∣ses of the Solids of these three∣sided figures above a C n•• Of the Solid of a three-si∣ded figure of 5 Means	30/24	24/30	18/36	12/42	6/48
The Exces∣ses of the Solids of these three∣sided figures above a C n•• Of the Solid of a three-si∣ded figure of 6 Means	36/27	30/33	24/39	18/45	12/51	6/57
The Exces∣ses of the Solids of these three∣sided figures above a C n•• Of the Solid of a three-si∣ded figure of •• Means	42/30	36/36	20/42	24/48	18/54	12/60	6/66

9 If any of these Deficient Figures, of which I have now spo∣ken, as A B C D (in the 5th figure) be inscribed within the Com∣plete figure B E, having A D C E for its Complement; and there be made upon C B produced, the triangle A B I; and the Parallelo∣gram A B I K be completed; and there be drawn parallel to the straight line C I, any number of lines as M F, cutting every one of them the crooked line of the Deficient Figure in D, and the straight lines A C, A B and A I in H, G and L; and as G F is to G D, so G L be made to another G N; and through all the points N there be drawn the line A N I, there will be a Deficient Figure A N I B, whose Complement will be A N I K. I say the figure A N I B is to the triangle A B I, as the Deficient Figure A B C D twice taken, is to the same Deficient Figure together with the Complete figure B E.

For as the proportion of A B to A G, that is, of G M to G L, is to the proportion of G M to G N; so is the magnitude of the fi∣gure A N I B, to that of its Complement A N I K (by the 2d. Art. of this Chapter.)

But (by the same Article), As the proportion of A B to A G, that is, of G M to G L, is to the proportion of G F to G D, that is, (by construction) of G L to G N; so is the figure A B C D to its Complement A D C E.

And by Composition, As the proportion of G M to G L, together with that of G L to G N, is to the proportion of G M to G L; so is the complete figure B E, to the Deficient Figure A B C D.

And by Conversion, As the proportion of G M to G L, is to both the proportions of G M to G L and of G L to G N, that is, to the proportion of G M to G N (which is the proportion compounded of both); so is the Deficient Figure A B C D, to the complete Fi∣gure B E.

But it was, As the proportion of G M to G L, to that of G M to G N; so the figure A N I B to its Complement A N I K. And therefore, A B C D. B E :: A N I B. A N I K are proportio∣nals. And by Composition, A B C D+B E. A B C D :: B K. A N I B are proportionals.

And by doubling the Consequents A B C D+B E. 2 A B C D :: B K. 2 A N I B are proportionals.

And by taking the halfes of the third & the fourth A B C D+B E. 2 A B C D :: A B I. A N I B are also proportionals; which was to be proved.

10 From what has been said of Deficient Figures described in a Parallelogram, may be found out what proportions Spaces transmitted with accelerated Motion in determined times, have to the times themselves, according as the moved Body is accelerated in the several times with one or more degrees of Velocity.

For, let the Parallelogram A B C D (in the 6th figure) and in it the three-sided figure D E B C be described; and let F G be drawn any where parallel to the base, cutting the Diagonal B D in H, and the crooked line B E D in E; & let the proportion of B C to B F be (for example) triplicate to that of F G to F E; whereupon the figure D E B C will be triple to its Complement B E D A; and in like manner, I F being drawn parallel to B C, the three-sided fi∣gure E K B F will be triple to its Complement B K E I. Where∣fore, the parts of the Deficient Figure cut off from the Vertex by straight lines parallel to the base, namely D E B C and E K B F, will be to one another as the Parallelograms A C and I F; that is, in proportion compounded of the proportions of the alti∣tudes and bases. Seeing therefore the proportion of the altitude B C to the altitude B F is triplicate to the proportion of the base D C to the base F E, the figure D E B C to the figure E K B F will be quadruplicate to the proportion of the same D C to F E. And by the same method, may be found out, what proportion any of the said three-sided figures, has to any part of the same cut off from the Vertex by a straight line parallel to the base.

Now as the said figures are understood to be described by the continual decreasing of the base, as of C D (for example) till it end in a point, as in B; so also they may be understood to be de∣scribed by the continual encreasing of a point, as of B, till it acquire any magnitude, as that of C D.

Suppose now the figure B E D C to be described by the encrea∣sing of the point B to the magnitude C D. Seeing therefore the propor ion of B C to B F is triplicate to that of C D to F E, the

proportion of F E to C D will by Conversion (as I shall presently demonstrate) be triplicate to that B F to B C. Wherefore, if the straight line B C be taken for the measure of the time in which the point B is moved, the Figure E K B F will represent the Sum of all the encreasing Velocities in the time B F; and the figure D E B C will in like manner represent the Summe of all the encreasing Velocities in the time B C. Seeing therefore the pro∣portion of the figure E K B F to the figure D E B C, is compound∣ed of the proportions of altitude to altitude, and base to base; and seeing the proportion of F E to C D is triplicate to that of B F to B C; the proportion of the figure E K B F to the figure D E B C, will be quadruplicate to that of B F to B C; that is, the proportion of the Sum of the Velocities in the time B F, to the Sum of the Veloci∣ties in the time B C wil be quadruplicate to the proportion of B F to B C. Wherfore if a Body be moved from B with Velocity so encrea∣sing, that the Velocity acquired in the time B F, be to the Velocity acquired in the time B C in triplicate proportion to that of the times themselves B F to B C, and the Body be carried to F in the time B F; the same Body in the time B C will be carried through a line equal to the fifth proportional from B F in the continual proportion of B F to B C. And by the same manner of working, we may determine, what Spaces are transmitted by Velocities en∣creasing according to any other proportions.

It remains, that I demonstrate the proportion of F E to C D, to be triplicate to that of B F to B C. Seeing therefore the pro∣portion of C D, that is of F G to F E is subtriplicate to that of B C to B F; the proportion of F G to F E will also be subtri∣plicate to that of F G to F H. Wherefore the proportion of F G to F H is triplicate to that of F G, that is, of C D to F E. But in four continual proportionals, of which the least is the first, the pro∣portion of the first to the fourth (by the 16 Art. of the 13 Chap.) is subtriplicate to the proportion of the third to the same fourh. Wherefore the proportion of F H to G F is subtriplicate to that of F E to C D; and therefore the proportion of F E to C D is triplicate to that of F H to F G, that is, of B F to B C, which was to be proved.

It may from hence be collected, that when the Velocity of a

Body, is encreased in the same proportion with that of the times, the degrees of Velocity above one another proceed as numbers do in immediate succession from Unity, namely, as 1, 2, 3, 4, &c. And when the Velocity is encreased in proportion duplicate to that of the times, the degrees proceed as numbers from Unity skipping One, as 1, 3, 5, 7, &c. Lastly, when the proportions of the Veloci∣ties are triplicate to those of the times, the progression of the de∣grees is as that of numbers from Unity skipping Two in every place, as 1, 4, 7, 10, &c. and so of other proportions. For Geome∣trical proportionals, when they are taken in every point, are the same with Arithmetical proportionals.

11 Moreover, it is to be noted, that as in quantities which are made by any magnitudes decreasing, the proportions of the figures to one another, are as the proportions of the altitudes to those of the bases; so also it is in those which are made with motion de∣creasing, which motion is nothing else but that power by which the described figures are greater or less. And therefore in the de∣scription of Archimedes his Spiral, which is done by the continual diminution of the Semidiameter of a Circle in the same proporti∣on in which the Circumference is diminished, the Space which is contained within the Semidiameter and the Spiral Line, is a third part of the whole Circle. For the Semidiameters of Circles, in as much as Circles are understood to be made up of the aggre∣gate of them, are so many Sectors; and therefore in the descripti∣on of a Spiral, the Sector which describes it, is diminished in du∣plicate proportions to the diminutions of the Circumference of the Circle in which it is inscribed; so that the Complement of the Spiral (that is, that space in the Circle which is without the Spi∣ral Line,) is double to the space within the Spiral Line. In the same manner, if there be taken a mean proportional every where be∣tween the Semidiameter of the Circle which contains the Spiral, and that part of the Semidiameter which is within the same, there will be made another figure, which will be half the Circle. And to conclude, this Rule serves for all such Spaces as may be described by a Line or Superficies decreasing either in magnitude or power; so that if the proportions in which they decrease, be com∣mensurable to the proportions of the times in which they de∣crease,

the magnitudes of the figures they describe will be known.

12 The truth of that proposition which I demonstrated in the second Article (which is the foundation of all that has been said concerning Deficient Figures) may be derived from the Elements of Philosophy, as having i•••• original in this; That all equality and in∣equality between two effects, (that is, all Proportion) proceeds from, and is determined by the equal and unequal causes of those effects, or from the pro∣portion which the causes concurring to one effect, have to the causes which con∣curre to the producing of the other effect; and that therefore the propor∣tions of Quantities are the same with the proportions of their cau∣ses. Seeing therefore two Deficient Figures (of which one is the Complement of the other) are made, one by motion decreasing in a certain time and proportion, the other by the loss of Motion in the same time, the causes which make and determine the quanti∣ties of both the figures, so, that they can be no other then they are, differ onely in this, that the proportions by which the quantity which generates the figure proceeds in describing of the same, (that is, the proportions of the remainders of all the times and altitudes) may be other proportions then those by which the same generating quantity decreases in making the Comple∣ment of that Figure, (that is, the proportions of the quantity which generates the Figure continually diminished.) Where∣fore, as the proportions of the quantity in which Motion is lost, is to that of the decreasing quantities by which the Deficient Figure is generated, so will the Defect or Complement be to the Figure it self which is generated.

13 There are also other quantities which are determinable from the knowledge of their causes, namely, from the comparison of the Motions by which they are made, and that more easily then from the common Elements of Geometry. For example, That the Superficies of any portion of a Sphere, is equal to that Circle, whose Radius is a straight Line drawn from the Pole of the porti∣on to the Circumference of its base, I may demonstrate in this manner. Let B A C (in the 7 Figure) be a portion of a Sphere, whose Axis is A E, & whose base is B C; & let A B be the straight line drawn from the Pole A to the base in B; and let A D, equal to A B, touch the great Circle B A C in the Pole A. It is to be proved

that the Circle made by the Radius A D, is equal to the Superfi∣cies of the portion B A C. Let the plain A E B D be understood to make a revolution about the Axis A E; & it is manifest that by the straight line A D a Circle will be described; and by the arch A B the Superficies of a portion of a Sphere, and lastly, by the Subtense A B the Superficies of a right Cone. Now seeing both the straight line A B, and the arch A B make one and the same revolution, and both of them have the same extreme points A and B, the cause why the the Spherical Superficies which is made by the arch, is greater then the Conical Superficies which is made by the Sub∣tense, is, that A B the arch, is greater then A B the Subtense; and the cause why it is greater consists in this, that although they be both drawn from A to B, yet the Subtense is drawn straight, but the arch angularly, namely according to that angle which the arch makes with the Subtense, which angle is equal to the angle D A B (for an angle of contingence adds nothing to an angle of a Seg∣ment, as has been shewn in the 14 Chapter at the 16th Article.) Wherefore the magnitude of the angle D A B is the cause why the Superficies of the portion described by the arch A B, is greater then the Superficies of the right Cone described by the Subtense A B.

Again, the cause why the Circle described by the Tangent A D is greater then the Superficies of the right Cone described by the Subtense A B (notwitstanding that the Tangent and the Subtense are equal, and both moved round in the same time) is this, that A D stands at right angles to the Axis, but A B obliquely; which obli∣quity consists in the same angle D A B. Seeing therefore the quan∣tity of the angle D A B is that which makes the excess both of the Superficies of the Portion, and of the Circle made by the Radius A D, above the superficies of the Right Cone described by the sub∣tense A B; it follows, that both the Superficies of the Portion, and that of the Circle, do equally exceed the Superficies of the Cone. Wherefore, the Circle made by A D, or A B, and the Spherical Superficies made by the arch A B, are equal to one another; which was to be proved.

••4 If these Deficient Figures which I have described in a 〈◊〉〈◊〉, were capable of exact description, then any number

of mean proportionals might be found out between two straight lines given. For example, in the Parallelogram A B C D, (in the 8th. Figure) let the three-sided figure of two Means be described, (which many call a Cubical Parabola); and let R and S be two given straight lines; between which, if it be required to find two mean proportionals, it may be done thus. Let it be as R to S, so B C to B F; and let F E be drawn parallel to B A, and cut the crooked line in E; then through E let G H be drawn parallel and equal to the straight line A D, and cut the Diagonal B D in I; for thus we have G I the greatest of two Means between G H and G E, as ap∣pears by the description of the figure in the 4th Article. Where∣fore if it be as G H to G I, so R to another line T, that T will be the greatest of two Means between R and S. And therefore if it be again as R to T, so T to another line X, that will be done which was required.

In the same manner, four mean proportionals may be found out, by the description of a three-sided figure of four Means; and so, any other number of Means, &c.

CHAP. XVIII. Of the Equation of Straight Lines with the Crooked Lines of Parabolas and other Figures made in imitation of Parabolas.

• 1 To find a straight Line equal to the crooked Line of a Semiparabola.
• 2 To find a straight Line equal to the Crooked Line of the first Semiparabola∣ster, or to the Crooked Line of any other of the Deficient Figures of the Ta∣ble of the 3d. Article of the pr••••edent Chapter.

1 AParabola being given, to find a Straight Line equal to the Crooked Line of the Semi∣parabola.

Let the Parabolical Line given be ABC (in the first Figure), and the Diameter found be AD, and the base drawn DC, and the Pa∣rallelogram ADCE being completed, draw the straight Line AC. Then dividing AD into two equal parts in F, draw FH equal and parallel to DC, cutting AC in K, and the parabolical line in O; and between FH and FO take a mean proportional FP, and draw AO, AP and PC. I say that the two Lines AP and PC taken together as one Line, is equal to the pa∣rabolical line ABOC.

For the line ABOC being a parabolical line, is generated by the concourse of two Motions, one Uniform from A to E, the o∣ther in the same time uniformly accelerated from rest in A to D. And because the motion from A to E is uniform, AE may re∣present the times of both those motions from the beginning to the end. Let therefore AE be the time; and consequently the lines

ordinately applyed in the Semiparabola, will designe the parts of time wherein the Body that describeth the line ABOC is in eve∣ry point of the same; so that as at the end of the time AE or DC it is in C, so at the end of the time FO it will be in O. And because the Velocity in AD is encreased uniformly, that is, in the same proportion with the time, the same lines ordinately applyed in the Semiparabola will designe also the continual augmentations of the Impetus, till it be at the greatest, designed by the base DC. There∣fore supposing Uniform motion in the line AF, in the time FK the Body in A by the concourse of the two uniform motions in AF and FK will be moved uniformly in the line AK; and KO wil be the encrease of the Impetus or Swiftness gained in the time FK; and the line AO will be uniformly described by the concourse of the two uniform motions in AF and FO in the time FO. From O draw OL parallel to EC, cutting AC in L; & draw LN parallel to DC, cutting EC in N, and the parabolical line in M; and produce it on the other side to AD in I; and IN, IM and IL will be (by the construction of a Parabola) in continual proportion, & equal to the three lines FH, FP and FO; and a straight line parallel to EC passing through M will fall on P; and therefore OP will be the encrease of Impetus gained in the time FO or IL. Lastly, produce PM to CD in Q; and QC, or MN, or PH will be the encrease of Impetus proportional to the time FP, or IM, or DQ. Suppose now uniform motion from H to C in the time PH. Seeing there∣fore in the time FP with uniform motion and the Impetus encrea∣sed in proportion to the times, is described the straight line AP; and in the rest of the time and Impetus, namely PH, is described the line CP uniformly; it followeth that the whole line APC is described with the whole Impetus, and in the same time where∣with is described the parabolicall line ABC; and therefore the line APC, made of the two straight lines AP and PC, is equal to the parabolical line ABC; which was to be proved.

2 To find a Straight line equal to the Crooked line of the first Semiparabolaster.

Let ABC be the Crooked line of the first Semiparabolaster; AD the Diameter; DC the Base; and let the Parallelogram

completed be ADCE, whose Diagonal is AC. Divide the Diameter into two equal parts in F, and draw FH equal and parallel to DC, ••utting AC in K, the Crooked line in O, and EC in H. Then draw OL parallel to EC, cutting AC in L; and draw LN parallel to the base DC, cutting the Crooked line in M, and the straight line EC in N; and produce it on the other side to AD in I. Lastly, through the point M draw PMQ parallel and equal to HC, cutting FH in P; and joyn CP, AP and AO. I say the two Straight lines AP and PC are equal to the Crooked line ABOC.

For the line ABOC being the Crooked line of the first Semi∣parabolaster, is generated by the concourse of two Motions, one uniform from A to E, the other in the same time accelerated from rest in A to D, so as that the Impetus encreaseth in proportion per∣petually triplicate to that of the encrease of the time, or (which is all one) the lengths transmitted are in proportion triplicate to that of the times of their transmission; for as the Impetus or Quicknesses encrease, so the Lengths transmitted encrease also. And because the motion from A to E is uniform, the line AE may serve to re∣present the time, and consequently the lines ordinately drawn in the Semiparabolaster, will designe the parts of time wherein the Body beginning from rest in A, describeth by its motion the Crook∣ed line ABOC. And because DC which represents the greatest acquired Impetus is equal to AE, the same ordinate lines will represent the several augmentations of the Impetus encreasing from rest in A. Therefore supposing uniform Motion from A to F in the time FK, there will be described by the concourse of the two uniform Motions AF and FK the line AK uniformly, and KO will be the encrease of Impetus in the time FK; And by the concourse of the two uniform Motions in AF and FO, will be described the line AO uniformly. Through the point L draw the straight line LMN parallel to DC, cutting the straight line AD in I, the crooked line ABC in M, and the straight line EC in N; and through the point M the straight line PMQ paral∣lel and equal to HC, cutting DC in Q, and FH in P. By the con∣course therefore of the two uniform Motions in AF and FP in the time FP will be uniformly described the straight line AP; and LM or OP will be the encrease of Impetus to be added for the time FO. And because the proportion if IN

to I L is triplicate to the proportion of I N to I M, the pro∣portion of F H to F O will also be triplicate to the propor∣tion of F H to F P; and the proportional Impetus gained in the time F P is P H. So that F H being equal to P C which de∣signed the whole Impetus acquired by the acceleration, there is no more encrease of Impetus to be computed. Now in the time P H suppose an uniform motion from H to C; and by the two uniform motions in C H and H P will be described uni∣formly the Straight line P C. Seeing therefore the two Straight lines A P and P C are described in the time A E with the same encrease of Impetus wherewith the Crooked line A B C is descri∣bed in the same time A E, that is, seeing the Line A P C and the Line A B C are transmitted by the same Body in the same Time, & with equal Velocities, the Lines themselves are equal; which was to be demonstrated.

By the same method, if any of the Semiparabolasters in the Table of the 3d Article of the precedent Chapter be exhibi∣ted, may be found a Straight line equal to the Crooked line there∣of, namely, by dividing the Diameter into two equal parts, and proceeding as before. Yet no man hitherto hath compared any Crooked with any Straight Line, though many Geometricians of every Age have endeavoured it. But the cause why they have not done it may be this, that there being in Euclide no Definition of E∣quality, nor any mark by which to judge of it besides Congruity (which is the 8th. Axiome of the first Book of his Elements) a thing of no use at all in the comparing of Straight and Crooked; and others after Euclide (except Archimedes and Apollonius, and in our time Bo••a••entura) thinking the industry of the Ancients had reach∣ed to all that was to be done in Geometry, thought also, that all that could be propounded, was either to be deduced from what they had written, or else that it was not at all to be done. It was therefore disputed by some of those Ancients themselves, whether there might be any Equality at all between Crooked and Straight Lines; Which question Archimedes (who assumed that some Straight line•• was equal to the Circumference of a Circle) seems to have despised, as he had reason. And there is a late Wri∣ter that granteth that between a Straight and a Crooked Line there is Equality; but now, now sayes he, since the fall of Adam, without the special assistance of Divine Grace, it is not to be found▪

CHAP. XIX. Of Angles of Incidence and Reflection, equal by supposition.

• 1 If two straight lines falling upon another straight line be parallel, the lines reflected from them shall also be parallel.
• 2 If two straight lines drawn from one point, fall upon another straight line, the lines reflected from them, if they be drawn out the other way, will me••t in an angle equal to the angle made by the lines of Incidence.
• 3 If two straight parallel lines drawn (not oppo∣sitely but) from the same parts, fall upon the Circumference of a Circle, the lines reflected from them, if produced they meet within the Circle, will make an angle double, to that which is made by two straight lines drawn from the Center to the points of Incidence.
• 4 If two straight lines drawn from the same point without a Circle fall upon the Circumference, and the lines reflected from them, being produced meet within the Circle, they will make an angle equal to twice that angle which is made by two straight lines drawn from the Center to the points of Incidence, together with the angle which the incident lines themselves make.
• 5 If two straight lines drawn from one point fall upon the concave Circumference of a Circle, and the angle they make be less then twice the angle at the Center, the lines reflected from them, and meeting within the Circle, will make an angle which being added to the angle of the incident lines, will be equal to twice the angle at the Center.
• 6 If through any one point two unequal Chords be drawn cutting one ano∣ther, and the Center of the Circle be not placed between them, and the lines reflected from them concurre wheresoever, there cannot through the point through which the two former lines were drawn, be drawn any other straight line, whose reflected line shall pass through the common point of the two for∣mer lines reflected.
• 7 In equal Chords the same is not true.
• 8 Two points being given in the Circumference of a Circle, to draw two straight lines to them, so as that their reflected lines may contain any angle given.
• 9 If a straight line falling upon the Circumference of a Circle be produced till it reach the Semidiameter, and that part of it which is intercepted be∣tween the Circumference and the Semidiameter, be equal to that part of the Semidiameter which is between the point of concourse & the center, the refle∣cted line will be parallel to the Semidiameter.
• 10 If from a point within a

• Circle, two straight lines be drawn to the Circumference, and their reflected lines meet in the Circumference of the same Circle, the angle made by the reflected lines, will be a third part of the angle made by the incident lines.

WHether a Body, falling upon the superficies of another Body and being reflected from it, do make equal angles at that superficies, it belongs not to this place to dispute, being a knowledge which depends upon the natural causes of Reflection; of which hither∣to nothing has been said, but shall be spoken of hereafter.

In this place therefore let it be supposed, that the angle of Inci∣dence is equal to the angle of Reflection, that our present search may be applyed not to the finding out of the causes, but some con∣sequences of the same.

I call an Angle of Incidence, that which is made between a straight line and another line (straight or crooked) upon which it falls, and which I call the Line Reflecting; and an Angle of Reflection equal to it, that which is made at the same point between the straight line which is reflected, and the line reflecting.

1 If two straight lines which fall upon another straight line be be parallel, their reflected lines shall be also parallel.

Let the two straight lines AB and CD (in the 1 figure) which fall upon the straight line EF, at the points B and D, be parallel; and let the lines reflected from them be BG and DH. I say BG and DH are also parallel.

For the angles ABE and CDE are equal by reason of the pa∣rallellelisme of AB and CD; and the angles GBF and HDF are equal to them by supposition; for the lines BG and DH are re∣flected from the lines AB and CD. Wherefore BG and DH are parallel.

2 If two straight lines drawn from the same point, fall upon another straight line, the lines reflected from them, if they be drawn out the other way, will meet in an angle equal to the angle of the Incident lines.

From the point AC (in the 2d. figure) let the two straight lines AB and AD be drawn; and let them fall upon the straight line EK at the points B and D; and let the lines BI and DG be refle∣cted from them. I say, IB and GD do converge, and that if they

be produced on the other side of the line EK they shall meet, as in F; and that the angle BFD shal be equal to the angle BAD.

For the angle of Reflection IBK is equal to the angle of Inci∣dence ABE; and to the angle IBK, its vertical angle EBF is e∣qual; and therefore the angle ABE is equal to the angle EBF. Again the angle ADE is equal to the angle of Reflection GDK, that is, to its vertical angle EDF; and therefore the two angles ABD and ADB of the triangle ABD, are one by one equal to the two angles FBD and FDB of the triangle FBD; Wherfore also the third angle BAD is equal to the third angle BFD, which was to be proved.

• Corollary 1. If the straight line AF be drawn, it will be perpen∣dicular to the straight line EK. For both the angles at E will be equal, by reason of the equality of the two angles ABE and FBE, and of the two sides AB and FB.
• Corollary 2. If upon any point between B and D there fall a straight line, as AC, whose reflected line is CH, this also pro∣duced beyond C, will fall upon F; which is evident by the demon∣stration above.

3 If from two points taken without a Circle, two straight pa∣rallel lines drawn (not oppositely but) from the same parts, fall up∣on the Circumference; the lines reflected from them, if produced they meet within the Circle, will make an angle double to that which is made by two straight lines drawn from the Center to the points of Incidence.

Let the two straight parallels AB and DC (in the 3d figure) fall upon the Circumference BC at the points B and C; and let the Center of the Circle be E; and let AB reflected be BF, and DC reflected be CG; and let the lines FB and GC produ∣ced meet within the Circle in H; and let EB and EC be conne∣cted. I say the angle FHG is double to the angle BEC.

For seeing AB and DC are parallels, and EB cuts AB in B, the same EB produced will cut DC somewhere; let it cut it in D, & let DC be produced howsoever to I, and let the intersection of DC & BF be at K. The angle therefore ICH (being external to the triangle CKH,) will be equal to the two opposite angles CKH and CHK. Again, ICE (being external to the triangle CDE)

is equal to the two angles at D and E. Wherefore the angle ICH, being double to the angle ICE, is equal to the angles at D and E twice taken; and therefore the two angles CKH and CHK are equal to the two angles at D and E twice taken. But the angle CKH is equal to the angles D and ABD, that is, D twice taken, (for AB and DC being parallels, the altern angles D, and ABD are equal). Wherefore CHK, that is the angle, FHG is also e∣qual to the angle at E twice taken; which was to be proved.

Corollary. If from two points taken within a circle, two straight parallels fall upon the circumference, the lines reflected from them shall meet in an angle, double to that which is made by two straight lines drawn from the center to the points of Incidence. For the parallels LB and IC falling upon the points B and C, are reflected in the lines BH and CH, and make the angle at H double to the angle at E, as was but now demonstrated-

4 If two straight lines drawn from the same point without a circle, fall upon the circumference, and the lines reflected from them being produced meet within the circle, they will make an angle equal to twice that angle which is made by two straight lines drawn from the center to the points of Incidence together with the angle which the incident lines themselves make.

Let the two straight lines AB and AC (in the 4th figure) be drawn from the point A to the circumference of the circle, whose center is D; and let the lines reflected from them be BE and CG, and being produced make within the circle the angle H; also let the two straight lines DB and DC be drawn from the center D to the points of Incidence B and C. I say the angle H is equal to twice the angle at D together with the angle at A.

For let AC be produced howsoever to I. Therefore the angle CH (which is external to the triangle CKH) will be equal to the two angles GKH and CHK. Again, the angle ICD (which is external to the triangle CLD) wil be equal to the two angles CLD and CDL. But the angle ICH is double to the angle ICD, and is therefore equal to the angles CLD and CDL twice taken. Wherefore the angles CKH and CHK are equal to the angles CLD and CDL twice taken. But the angle CLD (being exter∣nal

to the triangle ALB) is equal to the two angles LAB & LBA; & consequently CLD twice taken is equal to LAB & LAB twice taken. Wherefore CKH & CHK are equal to the angle CDL together with LAB and LBA twice taken. Also the angle CKH is equal to the angle LAB once, and ABK, that is, LBA twice taken. Wherefore the Angle CHK is equal to the remaining angle CDL (that is, to the angle at D) twice taken, and the angle LAB (that is, the angle at A) once taken; which was to be proved.

Corollary. If two straight converging lines, as IC and MB fall upon the concave circumference of a circle, their reflected lines, as CH and BH, will meet in the angle H, equal to twice the angle D, together with the angle at A made by the ••ncident lines produ∣ced. Or, if the Incident lines be HB and IC, whose reflected lines CH and BM meet in the point N, the angle CNB will be equal to twice the angle D, together with the angle CKH made by the lines of Incidence. For the angle CNB is equal to the angle H (that is, to twice the angle D) together with the two angles A and NBH (that is KBA). But the angles KBA and A are equal to the angle CKH. Wherefore the angle CNB is equal to twice the angle D, together with the angle CKH made by the lines of Incidence IC and HB produced to K.

5 If two straight lines drawn from one point, fall upon the con∣cave circumference of a circle, and the angle they make be lesse then twice the angle at the center; the lines reflected from them, and meeting within the circle, will make an angle, which being added to the angle of the incident lines, will be equal to twice the angle at the center.

Let the two Lines AB and AC (in the 5th figure) drawn from the point A, fall upon the concave circumference of the circle whose center is D; & let their reflected Lines BE and CE meet in the point E; also let the angle A be less then twice the angle D. I say the angles A and E together taken are equal to twice the angle D.

For let the straight Lines AB and EC cut the straight Lines DC and DB in the points G and H; and the angle BHC will be equal to the two angles EBH and E; also the same angle BHC will be equal to the two angles D and DCH; and in like manner the angle BGC will be equal to the two angles ACD & A, & the

same angle BGC will be also equal to the two angles DBG and D. Wherefore the four angles EBH, E, ACD and A are equal to the four angles D, DCH, DBG and D. If therefore equals be taken away on both sides, namely, on one side ACD and EBH, and on the other side DCH and DBG (for the angle EBH is e∣qual to the angle DBG, and the angle ACD equal to the angle DCH) the remainders on both sides will be equal, namely, on one side the angles A and E, and on the other the angle D twice taken. Wherefore the angles A and E are equal to twice the angle D.

Corollary. If the angle A be greater then twice the angle D, their reflected ••••ines will diverge. For, by the Corollary of the third Pro∣position, if the angle A be equal to twice the angle D, the reflect∣ed Lines BE and CE will be parallel; and if it be lesse, they will concurre, as has now been demonstrated; and therefore if it be greater, the reflected Lines BE and CE will diverge, and conse∣quently, if they be produced the other way, they will concurre, and make an angle equal to the excesse of the angle A above twice the angle D; as is evident by the fourth Article.

6 If through any one point, two unequal chords be drawn, cut∣ting one another, either within the circle, or (if they be produced) without it, and the center of the circle be not placed between them, and the Lines reflected from them concurre wheresoever; there cannot through the point through which the former Lines were drawn, be drawn another straight Line, whose reflected Line shall passe through the point where the two former reflected Lines con∣curre.

Let any two unequal chords, as BK and CH (in the 6th Figure) be drawn through the point A in the circle BC; and let their re∣flected Lines BD and CE meet in F; and let the center not be be∣tween AB and AC; and from the point A let any other straight Line as AG be drawn to the circumference between B and C. I say GN, which passes through the point F, where the reflected Lines BD and CE meet, will not be the reflected Line of AG.

For let the arch BL be taken equal to the arch BG, and the straight Line BM equal to the straight Line BA; and LM being drawn, let it be produced to the circūmference in O. Seeing there∣fore

BA and BM are equal, and the arch BL equal to the arch BG, and the angle MBL equal to the angle ABG, AG and ML will also be equal, and (producing GA to the circumference in I) the whole lines LO and GI will in like manner be equal. But LO is greater then GFN (as shall presently be demonstrated) and therefore also GI is greater then GN. Wherefore the angles NGC and IGB are not equal. Wherefore the Line GFN is not reflected from the Line of Incidence AG, and consequently no other straight Line (besides AB and AC) which is drawn through the point A, and fa••ls upon the circumference BC, can be reflect∣ed to the point F, which was to be demonstrated.

It remains that I prove LO to be greater then GN; which I shall do in this manner. LO and GN cut one another in P; and PL is greater then PG. Seeing now LP. PG :: PN. PO are proportionals, therefore the two Extremes LP and PO together taken, (that is LO), are greater then PG and PN together taken, (that is, GN,) which remained to be proved.

7 But if two equal chords be drawn through one point within a circle, and the Lines reflected from them meet in another point, then another straight Line may be drawn between them through the former point, whose reflected Line shall pass through the later point.

Let the two equal chords BC and ED (in the 7th figure) cut one another in the point A within the circle BCD; and let their reflected Lines CH and DI meet in the point F. Then dividing the arch CD equally in G, let the two chords GK and GL be drawn through the points A and F. I say GL will be the Line re∣flected from the chord KG. For the four chords BC, CH, ED and DI, are by supposition all equal to one another; and therefore the arch BCH is equal to the arch EDI; as also the angle BCH to the angle EDI; & the angle AMC to its vertical angle FMD; and the straight Line DM to the straight Line CM; and in like man∣ner, the straight Line AC to the straight Line FD; and the chords CG and GD being drawn, will also be equal; as also the angles FDG and ACG, in the equal Segments GDI and GCB. Wherefore the straight Lines FG and AG are equal; and there∣fore the angle FGD is equal to the angle AGC, that is, the an∣gle

of Incidence equal to the angle of Reflection. Wherefore the line GL is reflected from the incident Line KG; which was to be proved.

Corollary. By the very sight of the figure, it is manifest, that if G be not the middle point between C and D, the reflected Line GL will not pass through the point F.

8 Two points in the circumference of a circle being given, to draw two straight Lines to them, so as that their reflected Lines may be parallel, or contain any angle given.

In the circumference of the circle whose center is A (in the 8th. figure) let the two points B and C be given; and let it be required to draw to them from two points taken without the circle, two incident Lines, so, that their reflected Lines may (first) be parallel.

Let AB and AC be drawn; as also any incident Line DC, with its reflected Line CF; and let the angle ECD be made double to the angle A; and let HB be drawn parallel to EC, and produced till it meet with DC produced in I. Lastly, producing AB indefinitely to K, let GB be drawn, so, that the angle GBK may be equal to the angle HBK, and then GB will be the refle∣cted Line of the incident Line HB. I say DC and HB are two incident Lines, whose reflected Lines CF and BG are parallel.

For seeing the angle ECD is double to the angle BAC, the angle HIC is also (by reason of the parallels EC and HI) double to the same BAC; Therefore also FC and GB (namely the lines reflected from the incident lines DC and HB are parallel. Wherefore the first thing required, is done.

Secondly, let it be required to draw to the points B & C two straight lines of Incidence, so, that the lines reflected from them may contain the given angle Z.

To the angle ECD made at the point C, let there be added on one side the angle DCL equal to half Z, and on the other side the angle ECM equal to the angle DCL; and let the straight Line BN be drawn parallel to the straight line CM; and let the angle KBO be made equal to the angle NBK; which being done, BO will be the Line of Reflection from the Line of Incidence NB. Lastly, from the incident Line LC, let the reflected Line CO be drawn, cutting BO at O, and making the angle COB. I say the angle COB is equal to the angle Z.

Let NB be produced till it meet with the straight line LC pro∣duced in P. Seeing therefore the angle LCM is by construction equal to twice the angle BAC together with the angle Z; the angle NPL (which is equal to LCM by reason of the parallels NP and MC) will also be equal to twice the same angle BAC together with the angle Z. And seeing the two straight lines OC and OB fall from the point O upon the points C and B; and their reflected lines LC and NB meet in the point P; the angle NPL will be e∣qual to twice the angle BAC together with the angle COP. But I have already proved the angle NPL to be equal to twice the angle BAC together with the angle Z. Therefore the angle COP is e∣qual to the angle Z; Wherefore, Two points in the circumference of a Circle being given, I have drawn, &c. which was to be done.

But if it be required to draw the incident Lines from a point within the circle, so, that the Lines reflected from them may contain an angle equal to the angle Z, the same method is to be used, saving that in this case the angle Z is not to be added to twice the angle BAC, but to be taken from it.

9 If a straight line falling upon the circumference of a circle, be produced till it reach the Semidiameter, and that part of it which is intercepted between the circumference and the Semidia∣meter, be equal to that part of the Semidiameter which is between the point of concourse and the center, the reflected Line will be parallel to the Semidiameter.

Let any Line AB (in the 9th figure) be the Semidiameter of the circle whose center is A; and upon the circumference BD let the straight Line CD fall, and be produced till it cut AB in E, so, that ED and EA may be equal; & from the incident Line CD let the Line DF be reflected. I say AB and DF will be parallel.

Let AG be drawn through the point D. Seeing therefore ED and EA are equal, the angles EDA and EAD will also be e∣qual. But the angles FDG and EDA are equal (for each of them is half the angle EDH or FDC.) Wherefore the angles FDG and EAD are equal; and consequently DF and AB are parallel; which was to be proved.

Corollahy. If EA be greater then ED, then DF and AB being produced will concurre; but if EA be less then ED, then BA and DH being produced will concurre.

10 If from a point within a circle, two straight Lines be drawn to the Circumference, and their reflected Lines meet in the Cir∣cumference of the same circle, the angle made by the Lines of Re∣flection, will be a third part of the angle made by the Lines of In∣cidence.

From the point B (in the 10th figure) taken within the circle whose center is A, let the two straight lines BC and BD be drawn to the circumference; and let their reflected Lines CE and DE meet in the circumference of the same circle at the point E. I say the angle CED will be a third part of the angle CBD.

Let AC and AD be drawn. Seeing therefore the angles CED and CBD together taken, are equal to twice the angle CAD (as has been demonstrated in the 5th article); and the angle CAD twice taken is quadruple to the angle CED; the angles CED and CBD together taken, will also be equal to the angle CED four times taken; and therefore if the angle CED be taken away on both sides, there will remain the angle CBD on one side, equal to the angle CED thrice taken on the other side; which was to be demonstrated.

Coroll. Therefore a point being given within a Circle, there may be drawn two Lines from it to the Circumference, so as their refle∣cted Lines may meet in the Circumference. For it is but trisect∣ing the Angle CBD; which how it may be done, shall be shewn in the following Chapter.

CHAP. XX. Of the Dimension of a Circle, and the Division of Angles or Arches.

• 1 The Dimension of a Circle neer determined in Numbers by Archimedes and others.
• 2 The first attempt for the finding out of the Dimension of a Circle by Lines.
• 3 The second attempt for the finding out of the Dimen∣sion of a Circle from the consideration of the nature of Crookedness.
• 4 The third attempt; and some things propounded to be further searched into.
• 5 The Equation of the Spiral of Archimedes with a straight Line.
• 6 Of the Analysis of Geometricians by the Powers of Lines.

1 IN the comparing of an Arch of a Circle with a Straight Line, many and great Geometricians, even from the most ancient times, have exercised their wits; and more had done the same, if they had not seen their pains, though undertaken for the com∣mon good, if not brought to perfection, vilified by those that envy the prayses of other men. Amongst those Ancient Writers whose Works are come to our hands, Archimedes was the first that brought the Length of the Perimeter of a Circle within the limits of Numbers very litle differing from the truth; demonstrating the same to be less then three Diameters and a seventh part, but great∣er then three Diameters and ten seventy one parts of the Diame∣ter. So that supposing the Radius to consist of 10000000 equal parts, the Arch of a Quadrant will be between 15714285 and 15 04225 of the same parts. In our times Ludovicus Van Cullen & Willebrordus Snellius with joint endeavour have come yet neerer to the truth; and pronounced from true Principles, that the Arch of a Quadrant (putting, as before 10000000 for Radius) differs not one whole Unity from the number 15707963; which, if they had

exhibited their arithmetical operations (and no man had discove∣red any errour in that long work of theirs) had been demonstrated by them. This is the furthest progress that has been made by the way of Numbers; and they that have proceeded thus far de∣serve the praise of Industry. Nevertheless, if we consider the bene∣fit (which is the scope at which all Speculation should aime) the improvement they have made has been little, or none. For any or∣dinary man may much sooner, & more accurately find a Straight Line equal to the Perimeter of a Circle, and consequently square the Circle, by winding a small thred about a given Cylinder, then any Geometrician shall do the same by dividing the Radius into 10000000 equal parts. But though the length of the Circumfe∣rence were exactly set out, either by Numbers, or mechanically, or onely by chance, yet this would contribute no help at all towards the Section of Angles, unless happily these two Problemes, To di∣vide a given Angle according to any proportion assigned, and To finde a Straight Line equal to the Arch of a Circle, were reciprocal, and fol∣lowed one another. Seeing therefore the benefit proceeding from the knowledge of the Length of the Arch of a Quadrant, consists in this, that we may there by divide an Angle according to any pro∣portion, either accurately, or at least accurately enough for com∣mon use; and seeing this cannot be done by Arithmetick, I thought fit to attempt the same by Geometry; and in this Chapter to make trial whether it might not be performed by the drawing of Straight and Circular Lines.

2 Let the Square A B C D (in the first figure) be described; and with the Radii A B, B C and D C the three Arches B D, C A and A C; of which let the two B D and C A cut one another in E, and the two B D and A C in F. The Diagonals therefore B D and A C being drawn will cut one another in the center of the Square G, and the two Arches B D and C A into two equal parts in H and Y; and the Arch B H D will be trisected in F and E. Through the Center G let the two Straight Lines K G L and M G N be drawn parallel and equal to the sides of the Square A B and A D, cutting the four sides of the same Square in the points K, L, M and N; which being done, K L will pass through F, and M N through E. Then let O P be drawn parallel and equal to the side B C, cutting the Arch B F D in F, and the sides A B and D C

in O and P. Therefore O F will be the Sine of the arch B F, which is an arch of 30 degrees; and the same O F will be equal to half the Radius. Lastly, dividing the arch B F in the middle in Q, let R Q the Sine of the arch B Q be drawn and produced to S▪ so that Q S be equal to R Q, and consequently R S be equal to the chord of the arch B F; and let F S be drawn and produced to T in the side B C. I say, the Straight Line B T is equal to the Arch B F; and con∣sequently that B V the triple of B T is equal to the Arch of the Quadrant B F E D.

Let T F be produced till it meet the side B A produced in X; and dividing O F in the middle in Z, let. Q Z be drawn and produced till it meet with the side B A produced. Seeing therefore the Straight Lines R S and O F are parallel, and divided in the midst in Q and Z, Q Z produced will fall upon X, and X Z Q produced to the side B C will cut B T in the midst in α.

Upon the Straight line F Z the fourth part of the Radius A B let the equilateral triangle a Z F be constituted; & upon the center a, with the Radius a Z let the arch Z F be drawn; which arch Z F will therefore be equal to the arch Q F the half of the arch B F. Again, let the straight line Z O be cut in the midst in b, and the straight line b O in the midst in c; and let the bisection be continued in this manner till the last part O c be the least that can possibly be taken; and upon it, and all the rest of the parts equal to it into which the straight line O F may be cut, let so many equilateral triangles be understood to be constituted; of which let the last be d O c. If therefore upon the center d, with the Radius d O be drawn the arch O c, and upon the rest of the equal parts of the straight line O F be drawn in like manner so many equal arches, all those arches together taken will be equal to the whole arch B F; & the half of them, namely, those that are comprehended between O & Z, or between Z & F will be equal to the arch B Q or Q F and in summe, what part soever the straight line O c be of the straight line O F, the same part will the arch O c be of the arch O F, though both the arch and the chord be infinitely bisected. Now seeing the arch O c is more crooked then that part of the arch B F which is equal to it; and seeing also that the more the straight line X c is produced the more it diverges from the straight line X O, if

the points O and c be understood to be moved forwards with straight motion in X O and X c, the arch O •• will thereby be ex∣tended by little and little, till at the last it come some-where ••o have the same crookedness with that part of the arch B F which is equal to it. In like manner, if the straight line X b be drawn, and the point b be understood to be moved forwards at the same time, the arch c b will also by little and little be extended, till its crook∣edness come to be equal to the crookedness of that part of the arch B F which is equal to it. And the same will happen in all those smal equal arches which are described upon so many equal parts of the straight line O F. It is also manifest, that by straight motion in X O and X Z all those small arches will lie in the arch B F in the points B, Q and F. And though the same small e∣quall arches should not be coincident with the equall parts of the arch B F in all the other points thereof, yet certainly they will constitute two crooked lines, not onely equall to the two arches B Q and Q F and equally crooked, but also having their cavity towards the same parts; which how it should be, unlesse all those small arches should be coincident with the arch B F in all its points, is not imaginable. They are therefore coincident, and all the straight lines drawne from X & passing through the points of division of the straight line O F, will also divide the arch B F into the same proportions into which O F is divided.

Now seeing X b cuts off from the point B the fourth part of the arch B F, let that fourth part be B e; and let the Sine thereof f e be produced to F T in g, for so f e will be the fourth part of the straight line f g, because as O b is to O F, so is f e to f g. But B T is greater then f g; and therefore the same B T is greater then four Sines of the fourth part of the arch B F. And in like m••nner, if the arch B F be subdivided into any number of equal par•••• whatsoever, it may be proved that the straight line B T is greater then the Sine of one of those small arches so many times 〈◊〉〈◊〉 as ••here be parts made of the whole arch B F. Wherefore the ••traight line B T is not lesse then the Arch B F. But neither can it be greater, because if any straight line whatsoever, lesse then B T, be draw•• below B T parallel to it and terminated in the straight line•• X B and X T, it would cut the arch B F; and

so the Sine of some one of the parts of the arch B F taken so often as that small arch is found in the whole arch B F, would be great∣er then so many of the same arches; which is absurd. Wherefore the Straight line B T is equal to the Arch B F; & the Straight line B V equal to the Arch of the Quadrant B F D; and B V four times taken, equal to the Perimeter of the Circle described with the Radius A B. Also the Arch B F and the Straight line B T are every where divided into the same proportions; and consequently any given Angle, whether greater or less then B A F may be divided into any proportion given.

But the straight line B V (though its magnitude fall within the terms assigned by Archimedes) is found, if computed by the Canon of Sines, to be somwhat greater then that w^ch is exhibited by the Ludol∣phine numbers. Nevertheless, if in the place of B T, another straight line, though never so little less, be substituted, the division of An∣gles is immediatly lost, as may by any man be demonstrated by this very Scheme.

Howsoever, if any man think this my Straight line B V to be too great, yet, seeing the Arch and all the Parallels are every where so exactly divided, and B V comes so neer to the truth, I desire he would seach out the reason, Why (granting B V to be precisely true) the Arches cut off should not be equal.

But some man may yet ask the reason why the straight lines drawn from X through the equal parts of the arch B F should cut off in the Tangent B V so many straight lines equal to them, seeing the connected straight line X V passes not through the point D, but cuts the straight line A D produced in l; and consequently require some determination of this Probleme. Concerning which, I will say what I think to be the reason, namely, that whilest the mag∣nitude of the Arch doth not exceed the magnitude of the Radius, that is, the magnitude of the Tangent B C, both the Arch and the Tangent are cut alike by the straight lines drawn from X; other∣wise not. For A V being connected, cutting the arch B H D in I, if X C being drawn should cut the same arch in the same point I, it would be as true that the Arch B I is equal to the Radius B C, as it is true that the Arch B F is equal to the straight line B T, and drawing X K it would cut the arch B I in the midst in i; Also

drawing A i and producing it to the Tangent B C in k, the straight line B k will be the Tangent of the arch B i, (which arch is equal to half the Radius) and the same straight line B k will be equal to the straight line k I. I say all this is true, if the preceding demonstrati∣on be true; and consequently the proportional section of the Arch and its Tangent proceeds hitherto. But it is manifest by the Gol∣den Rule, that taking B h double to B T, the line X h shall not cut off the arch B E which is double to the arch B F, but a much great∣er. For the magnitude of the straight lines X M, X B and M E be∣ing known (in numbers) the magnitude of the straight line cut off in the Tangent by the straight line X E produced to the Tangent may also be known; and it will be found to be less then B h; Wher∣fore the straight line Xh being drawn will cut off a part of the arch of the Quadrant greater then the arch B E. But I shall speak more fully in the next Article concerning the magnitude of the arch B I.

And let this be the first attempt for the finding out of the di∣mension of a Circle by the Section of the arch B F.

3 I shall now attempt the same by arguments drawn from the nature of the Crookedness of the Circle it self; but I shall first set down some Premisses necessary for this speculation; and

First, If a Straight line be bowed into an Arch of a Circle equal to it, as when a stretched thred which toucheth a Right Cylin∣der, is so bowed in every point, that it be every where coincident with the Perimeter of the base of the Cylinder, the Flexion of that line will be equal, in all its points; and consequently the Croo∣kedness of the Arch of a Circle is every where Uniform; which needs no other demonstration then this, That the Perimeter of a Circle is an Uniform line.

Secondly, and consequently, If two unequal Arches of the same Circle be made by the bowing of two straight lines equal to them, the Flexion of the longer line (whilest it is bowed into the greater Arch) is greater then the Flexion of the shorter line (whilest it is bowed into the lesser Arch) according to the proportion of the Ar∣ches themselves; and consequently, the Crookedness of the greater Arch is to the Crookedness of the lesser Arch; as the greater Arch is to the lesser Arch.

Thirdly, If two unequal Circles and a straight line touch one

another in the same point, the Crookedness of any Arch taken in the lesser Circle, will be greater then the Crookedness of an Arch equal to it taken in the greater Circle, in reciprocal proportion to that of the Radii with which the Circles are described; or, which is all one, any straight line being drawn from the point of Contact till it cut both the circumferences, as the part of that straight line cut off by the circumference of the greater Circle to that part which is cut off by the circumference of the lesser Circle.

For let A B and A C (in the second figure) be two Circles, tou∣ching one another and the straight line A D in the point A; and let their Centers be E and F; and let it be supposed, that as A E is to A F, so is the Arch A B to the Arch A H. I say the Crookedness of the Arch A C is to the Crookedness of the Arch A H, as A E is to A F. For let the straight line A D be supposed to be equal to the Arch A B, and the straight line A G to the Arch A C; and let A D (for example) be double to A G. Therefore by reason of the likeness of the Arches A B and A C, the straight line A B will be double to the straight line A C, and the Radius A E double to the Radius A F, and the Arch A B double to the Arch A H. And because the straight line A D is so bow∣ed to be coincident with the Arch A B equal to it, as the straight line A G is bowed to be coincident with the Arch A C e∣qual also to it, the Flexion of the straight line A G into the Crooked line A C will be equal to the Flexion of the straight Line A D into the Crooked line A B. But the Flexion of the straight line A D into the Crooked line A B is double to the the Flexion of the straight line A G into the Crooked line A H; and therefore the Flexion of the straight line A G into the Crook∣ed line A C is double to the Flexion of the same straight line A G into the Crooked line A H. Wherefore, as the Arch A B is to the Arch A C or A H; or as the Radius A E is to the Radius A F; or as the Chord A B is to the Chord A C; so reciprocally is the Flexion or Uniform Crookedness of the Arch A C, to the Flexion or Uni∣form Crookedness of the Arch A H, namely, here double. And this may by the same method be demonstrated in Circles whose Peri∣meters are to one another triple, quadruple, or in whatsoever gi∣ven proportion. The Crookedness therefore of two equal Arches

taken in several Circles are in proportion reciprocall to that of their Radii, or like Arches, or like Chords; which was to be de∣monstrated.

Let the Square A B C D be again described (in the third Figure,) and in it the Quadrants A B D, B C A and D A C; and dividing each side of the Square A B C D in the midst in E, F, G and H, let E G and F H be connected, which will cut one ano∣ther in the center of the Square at I, and divide the arch of the Quadrant A B D into three equal parts in K and L. Also the Dia∣gonals A C and B D being drawn will cut one another in I, and di∣vide the arches B K D and C L A into two equal parts in M and N. Then with the Radius B F let the arch F E be drawn, cutting the Diagonal B D in O; and dividing the arch B M in the midst in P, let the straight line E a equal to the chord B P be set off from the point E in the arch E F, and let the arch a b be taken equal to the arch O a, and let B a and B b be drawn and produced to the arch A N in c and d; and lastly, let the straight line A d be drawn. I say the Straight line A d is equal to the Arch A N or B M.

I have proved in the preceding article, that the arch E O is twice as crooked as the arch B P, that is to say, that the arch E O is so much more crooked then the arch B P, as the arch B P is more crooked then the straight line E a. The crookedness therefore of the chord E a, of the arch B P, and of the arch E O are as 0, 1, 2. Also the difference between the arches E O and E O, the difference between the arches E O and E a, and the difference between the arches E O and E b are as 0, 1, 2. So also the difference between the arches A N and A N, the difference between the arches A N and A c, and the difference between the arches A N and A d are as 0, 1, 2; and the straight line A c is double to the chord B P or E a, and the straight line A d double to the chord E b.

Again, let the straight line B F be divided in the midst in Q, and the arch B P in the midst in R; and describing the Quadrant B Q S (whose arch Q S is a fourth part of the arch of the Qua∣drant B M D, as the arch B R is a fourth part of the arch B M which is the arch of the Semiquadrant A B M) let the chord S e equal to the chord B R be set off from the point S in the arch S Q; and let B e be drawn and produced to the arch A N in f; which being done, the straight line A f will be quadruple to the

chord B R or S e. And seeing the crookedness of the arch S e or of the arch A c is double to the crookedness of the arch B R, the excess of the crookedness of the arch A f above the crookedness of the arch A c will be subduple to the ex∣cess of the crookedness of the arch A c above the crookedness of the arch A N; and therefore the arch N c will be double to the arch c f. Wherefore the arch c d is divided in the midst in f, and the arch N f is ¾ of the arch N d. And in like manner if the arch B R be bisected in V, and the straight Line B Q in X, and the quadrant B X Y be described, and the straight Line Y g equal to the chord B V be set off from the point Y in the arch Y X, it may be demonstrated that the straight Line B g being drawn and pro∣duced to the arch A N will cut the arch f d into two equal parts, and that a straight Line drawn from A to the point of that Se∣ction, will be equal to eight chords of the arch B V, and so on per∣petually; and consequently, that the straight Line A d is equal to so many equal chords of equal parts of the arch B M, as may be made by infinite bisections. Wherefore the Straight Line A d is equal to the Arch B M or A N, that is, to half the Arch of the Qua∣drant A B D or B C A.

Corollary. An Arch being given not greater then the arch of a Quadrant (for being made greater it comes again towards the Radius B A produced, from which it receded before) if a straight Line double to the chord of half the given arch be adapted from the beginning of the arch, and by how much the arch that is sub∣tended by it is greater then the given arch, by so much a greater arch be subtended by another straight Line, this Straight Line shall be equal to the first given Arch.

Supposing the Straight Line B V (in the first Figure) be equal to the arch of the Quadrant B H D, and A V be connected cutting the arch B H D in I, it may be asked what pro∣portion the arch B I has to the arch I D. Let therefore the arch A Y be divided in the midst in o, and in the straight line A D let A p be taken equal, and A q double to the drawn chord A o. Then upon the center A, with the Radius A q let an arch of a cir∣cle be drawne cutting the arch A Y in r, and let the arch Y r be doubled at t; which being done, the drawne straight line A t (by what has been last demonstrated) will be equall to the arch A Y.

Again, upon the Center A with the Radius A t let the arch tu be drawne cutting AD in u; and the straight line A u will be e∣quall to the arch AY. From the point u let the straight line us be drawn equal and parallell to the straight line AB, cutting MN in x, and bisected by MN in the same point x. Therefore the straight line A x being drawn and produced till it meet with BC produced in V, it will cut off BV double to B s, that is, equal to the arch BHD. Now let the point where the straight line AV cuts the arch BHD, be I; and let the arch DI be divided in the midst in y; and in the straight line DC, let D z be taken equal, and D δ double to the drawn chord D y; and upon the center D with the Radius D δ let an arch of a circle be drawn cutting the arch BHD in the point n; and let the arch nm be taken equal to the arch I n; which being done, the straight line D m will (by the last foregoing Corollary) be equal to the arch DI. If now the straight lines D m and CV be equal, the arch BI will be equal to the Radius AB or BC; and consequently XC being drawn will pass through the point I. Moreover, if the semicircle BHD β being completed, the straight lines β I and BI be drawn making a right angle (in the Semicircle) at I, and the arch BI be divided in the midst at i, it will follow that A i being connected will be parallel to the straight line β I, and being produced to BC in k, will cut off the straight line B k equal to the straight line k I, and equal also to the straight line Aγ cut off in AD by the straight line β I. All which is manifest, supposing the arch BI and the Radius BC to be equal.

But that the arch BI and the Radius BC are precisely equal, cannot (how true soever it he) be demonstrated, unless that be first proved w^ch is contained in the first article, namely, that the straight lines drawn from X through the equal parts of OF (produced to a certain length) cut off so many parts also in the Tangent BC seve∣rally equal to the several arches cut off; which they do most exactly as far as BC in the Tangent, and BI in the arch BE; in so much that no inequality between the arch BI and the Radius BC can be dis∣covered either by the hand or by ratiocination. It is therefore to be further enquired, whether the straight line AV cut the arch of the Quadrant in I in the same proportion as the point C divides

the straight line BV which is equal to the arch of the Quadrant. But however this be, it has been demonstrated that the straight line BV is equal to the arch BHD.

4 I shall now attempt the same dimension of a Circle another way, assuming the two following Lemma's.

Lemma 1. If to the Arch of a Quadrant, and the Radius, there be taken in continual proportion a third Line Z; then the Arch of the Semiquadrant, Half the chord of the Quadrant, and Z will also be in continual proportion.

For seeing the Radius is a mean proportional between the Chord of a Quadrant and its Semichord, and the same Radius a mean proportional between the Arch of the Quadrant and Z, the Square of the Radius will be equal as well to the Rectangle made of the Chord and Semichord of the Quadrant, as to the Rectangle made of the Arch of the Quadrant and Z; and these two Rectangles will be equal to one another. Wherfore, as the Arch of a Quadrant is to its Chord, so reciprocally is half the Chord of the Quadrant to Z. But as the Arch of the Quadrant is to its Chord, so is half the Arch of the Quadrant to half the Chord of the Quadrant. Where∣fore, as half the arch of the Quadrant is to half the Chord of the Quadrant (or to the Sine of 45 degrees) so is half the Chord of the Quadrant to Z; which was to be proved.

Lemma 2. The Radius, the Arch of the Semiquadrant, the Sine of 45 degrees, and the Semiradius are proportional.

For seeing the Sine of 45 degrees is a mean proportional be∣tween the Radius and the Semiradius; and the same Sine of 45 de∣grees is also a mean proportional (by the precedent Lemma) be∣tween the Arch of 45 degrees and Z; the Square of the Sine of 45 degrees will be equal as well to the Rectangle made of the Radius and Semiradius, as to the Rectangle made of the Arch of 45 de∣grees and Z. Wherefore, as the Radius is to the Arch of 45 de∣grees, so reciprocally is Z to the Semiradius; which was to be de∣monstrated.

Let now ABCD (in the fourth Figure) be a Square; and with the Radii AB, BC and DA let the three Quadrants ABD, BCA and DAC be described; and let the straight lines EF and GH drawn parallel to the Sides BC & AB, divide the Square ABCD into foure equal Squares. They will therefore cut the arch of the

Quadrant ABD into three equal parts in I and K, and the arch of the Quadrant BCA into three equal parts in K and L. Also let the Diagonals AC and BD be drawn, cutting the arches BID and ALC in M and N. Then upon the center H with the Radius HF equal to half the Chord of the arch BMD, or to the Sine of 45 degrees, let the arch FO be drawn cutting the arc•• CK in O; and let AO be drawn and produced till it meet with BC produ∣ced in P; also let it cut the arch BMD in Q, and the straight line DC in R. If now the straight line H Q be equal to the straight line DR, and being produced to DC in S cut off DS equal to half the straight line BP; I say then the Straight Line BP will be e∣qual to the Arch BMD.

For seeing PBA and ADR are like triangles, it will be as PB to the Radius BA or AD, so AD to DR; and therefore as well PB, AD and DR, as PB, AD (or A Q) and Q H are in continu∣all proportion; and producing HO to DC in T, DT will be equal to the Sine of 45 degrees, as shall by and by be demonstrated. Now DS, DT and DR are in continual porportion by the first Lemma; and by the second Lemma DC. DS:: DR. DF are proportionals. And thus it will be, whether BP be equal or not equal to the arch of the Quadrant BMD. But if they be equal, it will then be, as that part of the arch BMD which is equal to the Radius, is to the remainder of the same arch BMD; so A Q to H Q, or so BC to CP. And then will BP and the arch BMD be equal. But it is not demonstrated that the Straight Lines H Q and DR are equal; though if from the point B there be drawn (by the construction of the first figure) a Straight Line equal to the arch BMD, then DR to H Q, and also the half of the Straight Line BP to DS, will always be so equal, that no inequa∣lity can be discovered between them. I will therefore leave this to be further searched into. For though it be almost out of doubt, that the Straight Line BP and the arch BMD are equal, yet that may not be received without demonstration; and means of Demon∣stration the Circular Line admitteth none that is not grounded upon the nature of Flexion, or of Angles. But by that way I have already exhibited a Straight Line equal to the Arch of a Qua∣drant in the First and Second aggression.

It remains that I prove DT to be equal to the Sine of 45 de∣grees.

In BA produced let AV he taken equal to the Sine of 45 de∣grees; and drawing and producing VH, it will cut the arch of the Quadrant CNA in the midst in N, and the same arch again in O, and the Straight line DC in T, so, that DT will be equal to the Sine of 45 degrees, or to the straight line AV; also the Straight line VH will be equal to the straight line HI or the Sine of 60 de∣grees.

For the square of AV is equal to two squares of the Semiradius; and consequently the square of VH is equal to three Squares of the Semiradius. But HI is a mean proportional between the Semi∣radius and three Semiradii; and therefore the square of HI is e∣qual to three Squares of the Semiradius. Wherefore HI is eqval to HV. But because AD is cut in the midst in H, therefore VH and HT are equal; and therefore also DT is equal to the Sine of 45 degrees. In the Radius BA let BX be taken equal to the Sine of 45 degrees; for so VX will be equal to the Radius; and it will be as VA to AH the Semiradius, so VX the Radius to XN the Sine of 45 degrees. Wherefore VH produced passes through N. Lastly, upon the center V with the Radius VA let the arch of a circle be drawn cutting VH in Y; which being done, VY will be e∣qual to HO (for HO is by construction equal to the Sine of 45 de∣grees) and YH will be equal to OT; & therefore VT passes through O. All which was to be demonstrated.

I will here add certain Problemes, of which if any Analyst can make the construction, he will thereby be able to judge clearly of what I have now said concerning the dimension of a Circle. Now these Problems are nothing else (at least to sense) but certain symptomes accompanying the construction of the first and third fi∣gure of this Chapter.

Describing therefore again the Square ABCD (in the fifth fi∣gure) and the three Quadrants ABD, BCA and DAC, let the Diagonals AC & BD be drawn, cutting the arches BHD & CIA in the middle in H and I; & the straight lines EF and GL, divi∣ding the square ABCD into four equal squares, and trisecting the arches BHD and CIA, namely, BHD in K and M, and CIA

in M and O. Then dividing the arch BK in the midst in P, let QP the Sine of the arch BP be drawn and produced to R, so that QR be double to QP; and connecting KR, let it be produced one way to BC in S, and the other way to BA produced in T. Also let BV be made triple to BS, and consequently (by the second article of this Chapter) equall to the arch BD. This construction is the same with that of the first figure, which I thought fit to renew discharged of all lines but such as are necessary for my present purpose.

In the first place therefore, if AV be drawn, cutting the arch BHD in X, and the side DC in Z, I desire some Analyst would (if he can) give a reason, Why the straight lines TE and TC should cut the arch BD the one in Y, the other in X, so as to make the arch BY equal to the arch YX; or if they be not equal, that he would determine their difference.

Secondly, if in the side DA, the straight line Da be taken equal to DZ, and Va be drawn; Why Va and VB should be equal; or if they be not equal, What is the difference.

Thirdly, drawing Zb parallel and equal to the side CB, cutting the arch BHD in c, and drawing the straight line Ac, and produ∣cing it to BV in d; Why Ad should be equal and parallel to the straight line aV, and consequently equal also to the arch BD.

Fourthly, drawing eK the Sine of the arch BK, & taking (in eA produced) ef equal to the Diagonal AC, and connecting fC; Why fC should pass through a (which point being given, the length of the arch BHD is also given) and c; and why fe and fc should be equal; or if not, why unequal.

Fifthly, drawing fZ, I desire he would shew, Why it is equal to BV, or to the arch BD; or if they be not equal, What is their dif∣ference.

Sixtly, granting fZ to be equal to the arch BD, I desire he would determine whether it fall all without the arch BCA, or cut the same; or touch it, and in what point.

Seventhly, the Semicircle BDg being completed; Why gI be∣ing drawn and produced, should pass through X (by which point X the length of the arch BD is determined). And the same gI be∣ing yet further produced to DC in h; Why Ad (which is equal

to the arch BD) should pass through that point h.

Eighthly, upon the Center of the square ABCD, which let be k, the arch of the quadrant EiL being drawn, cutting eK produ∣ced in i; Why the drawn straight line iX should be parallel to the side CD.

Ninthly, in the sides BA and BC taking Bl and Bm severally equal to half BV, or to the arch BH, and drawing mn parallel and equal to the side BA, cutting the arch BD in o; Why the straight line wich connects Vl should pass through the point o,

Tenthly, I would know of him, Why the straight line which connects aH should be equal to Bm; or if not, how much it differs from it.

The Analyst that can solve these Problemes without knowing first the length of the arch BD, or using any other known Method then that which proceeds by perpetual bisection of an angle, or is drawn from the consideration of the nature of Flexion, shall do more then ordinary Geometry is able to perform. But if the Di∣mension of a Circle cannot be found by any other Method; then I have either found it, or it is not at all to be found.

From the known Length of the Arch of a Quadrant, and from the proportional Division of the Arch and of the Tangent BC, may be deduced the Section of an Angle into any given proportion; as also the Squaring of the Circle, the Squaring of a given Sector, and many the like propositions, which it is not necessary here to de∣monstrate. I will therefore onely exhibit a Straight line equal to the Spiral of Archimedes, and so dismiss this speculation.

5 The length of the Perimeter of a Circle being found, that Straight line is also found, which touches a Spiral at the end of its first conversion. For upon the center A (in the sixth figure) let the circle BCDE be described; and in it let Archimedes his Spiral AFGHB be drawn, beginning at A and ending at B. Through the center A let the straight line CE be drawn, cutting the Diameter BD at right angles; and let it be produced to I, so, that AI be equal to the Perimeter BCDEB. Therefore IB be∣ing drawn will touch the Spiral AFGHB in B; which is demon∣strated by Archimedes in his book de Spiralibus.

And for a Straight Line equal to the given Spiral AFGHB, it may be found thus.

Let the straight line AI (which is equal to the Perimeter BCDE) be bisected in K; and taking KL equal to the Radius AB, let the rectangle IL be completed. Let ML be understood to be the axis, and KL the base of a Parabola, and let MK be the crooked line thereof. Now if the point M be conceived to be so moved by the concourse of two movents, the one frō IM to KL with velocity encreasing continually in the same proportion with the Times, the other from ML to IK uniformly, that both those moti∣ons begin together in M and end in K; Galilaeus has demonstrated that by such motion of the point M, the crooked line of a Parabola will be described. Again, if the point A be conceived to be moved uniformly in the straight line AB, and in the same time to be car∣ried round upon the center A by the circular motion of all the points between A and B; Archimedes has demonstrated that by such motion will be described a Spiral line. And seeing the circles of all these motions are concentrick in A; and the interiour circle is alwayes lesse then the exteriour in the proportion of the times in which AB is passed over with uniform motion; the velocity also of the circular motion of the point A, will continually encrease proportionally to the times. And thus far the generations of the Pa∣rabolical line MK, and of the Spiral line AFGHB, are like. But the Uniform motion in AB concurring with circular motion in the Perimeters of all the concentrick circles, describes that circle, whose center is A, and Perimeter BCDE; and therefore that circle is (by the Coroll. of the first article of the 16 Chapter) the aggregate of all the Velocities together taken of the point A whilst it describes the Spiral AFGHB. Also the rectangle IKLM is the aggregate of all the Velocities together taken of the point M, whilest it describes the crooked line MK. And therefore the whole velocity, by which the Parabolicall line MK is described▪ is to the whole velocity with which the Spiral line AFGHB is described in the same time, as the rectangle IKLM, is to the Circle BCDE, that is to the triangle AIB. But because AI is bisected in K & the straight lines IM & AB are equal, therefore the rectangle IKLM and the triangle AIB are also equal. Wherefore the Spiral line AFGHB, and the Parabolical line MK, being described with equal velocity and in equal times, are equal to one another. Now in the first article of the 18 Chapter a straight line is found

out equal to any Parabolical line. Wherefore also a Straight line is found out, equal to a given Spiral line of the first revolution descri∣bed by Archimedes; which was to be done.

6 In the sixth Chapter, which is of Method, that which I should there have spoken of the Analyticks of Geometricians, I thought fit to deferre, because I could not there have been understood, as not having then so much as named Lines, Superficies, Solids, Equal and Unequal &c. Wherefore I will in this place set down my thoughts concerning it.

Analysis, is continual Reasoning from the Definitions of the terms of a proposition we suppose true, and again from the Definitions of the terms of those Definitions, and so on, till we come to some things known, the Composition whereof is the demonstration of the truth or falsity of the first suppo∣sition; and this Composition or Demonstration is that we call Synthesis. Analytica therefore is that art, by which our reason pro∣ceeds from something supposed, to Principles, that is, to prime Propositions, or to such as are known by these, till we have so ma∣ny known Propositions as are sufficient for the demonstration of the truth or falsity of the thing supposed. Synthetica is the art it self of Demonstration. Synthesis therefore and Analysis differ in nothing, but in proceeding forwards or backwards; and Logistica compre∣hends both. So that in the Analysis or Synthesis of any question, that is to say, of any Probleme, the Terms of all the Propositions ought to be convertible; or if they be enunciated Hypothetically, the truth of the Consequent ought not onely to follow out of the truth of its Antecedent, but contrarily also the truth of the Antecedent must necessarily be inferred from the truth of the Consequent. For otherwise, when by Resolution we are arrived at Principles, we cannot by Composition return directly back to the thing sought for. For those Terms which are the first in Analysis, will be the last in Synthesis; as for example, when in Resol••ing, we say, these two Rectangles are equal and therefore their sides are reciprocally proportional, we must necessarily in Compounding say, the sides of these Rectangles are reciprocally proportional and therefore the Rectangles themselves are equal; Which we could not say, •…•…ss Rectangles have their sides reciprocally proportional, and Rectangles are e∣qual, were Terms convertible.

Now in every Analysis, that which is sought, is the Proportion of two quantities; by which proportion (a figure being described) the quantity sought for may be exposed to Sense. And this Exposition is the end and Solution of the question, or the construction of the Probleme.

And seeing Analysis is reasoning from something supposed, till we come to Principles, that is, to Definitions, or to Theoremes for∣merly known; and seeing the same reasoning tends in the last place to some Equation; we can therefore make no end of Resolving, till we come at last to the causes themselves of Equality and Inequali∣ty, or to Theoremes formerly demonstrated from those causes; and so have a sufficient number of those Theoremes for the de∣monstration of the thing sought for.

And seeing also, that the end of the Analyticks, is either the con∣struction of such a Probleme as is possible, or the detection of the impossibility thereof; whensoever the Probleme may be solved, the Analyst must not stay, till he come to those things which contain the efficient cause of that whereof he is to make construction. But he must of necessity stay when he comes to prime Propositions; and these are Definitions. These Definitions therefore must contain the efficient cause of his Construction; I say of his Construction, not of the Conclusion which he demonstrates; for the cause of the Conclusion is contained in the premised propositions; that is to say, the truth of the proposition he proves, is drawn from the pro∣positions which prove the same. But the cause of his constructi∣on is in the things themselves, and consists in motion, or in the concourse of motions. Wherefore those propositions in which Analysis ends, are Definitions, but such, as signifie in what manner the construction, or generation of the thing proceeds. For other∣wise, when he goes back by Synthesis to the proofe of his Probleme, he will come to no Demonstration at all; there being no true De∣monstration but such as is scientificall; and no Demonstration is scientifical but that which proceeds from the knowledge of the causes from which the construction of the Probleme is drawne. To collect therefore what has been said into few words; ANALY∣SIS is Ratiocination from the supposed construction or generation of a thing to the efficient cause, or coefficient causes of that which is constructed or generated. And SYNTHESIS is Ratiocination from the first cau∣ses

of the Construction, continued through all the middle causes till we come to the thing it selfe which is constructed or generated.

But because there are many means by which the same thing may be generated, or the same Probleme be constructed, there∣fore neither do all Geometricians, nor doth the same Geometri∣cian alwayes use one and the same Method. For if to a certain quantity given, it be required to construct another quantity equal, there may be some that will enquire whether this may not be done by means of some motion. For there are quantities, whose equali∣ty and inequality may be argued from Motion and Time, as well as from Congruence; and there is motion, by which two quanti∣ties, whether Lines or Superficies, though one of them be crook∣ed, the other straight, may be made congruous or coincident. And this method Archimedes made use of in his Book de Spiralibus. Also the equality or inequality of two quantities may be found out and demonstrated from the consideration of Waight, as the same Archimedes did in his Quadrature of the Parabola. Besides, equality and equality are found out often by the division of the two quanti∣tyes into parts which are considered as undivisible; as Cavallerius Bonaventura has done in our time, and Archimedes often. Lastly, the same is performed by the consideration of the Powers of lines, or the roots of those Powers, and by the multiplication, division, addition and substraction, as also by the extraction of the roots of those Powers, or by finding where straight lines of the same pro∣portion terminate. For example, when any number of straight lines, how many soever, are drawne from a straight line, and passe all through the same point, looke what proportion they have, and if their parts continued from the point retaine every where the same proportion, they shall all terminate in a straight line. And the same happens if the point be taken between two Circles. So that the places of all their points of termination make either straight lines, or circumferences of Circles, and are called Plain Places. So also when straight parallel lines are applyed to one straight line, if the parts of the straight line to which they are applyed be to one another in proportion dupli∣cate to that of the contiguous applyed lines, they will all termi∣nate in a Conical Section; which Section being the place of their

termination, is called a Solid Place, because it serves for the find∣ing out of the quantity of any Equation which consists of three di∣mensions. There are therfore three ways of finding out the cause of Equality or Inequality between two given quantities; namely, First by the Computation of Motions (for by equal Motion, & equal Time equal Spaces are described,) and Ponderation is motion. Se∣condly By Indivisibles; because all the parts together taken are equal to the whole. And thirdly by the Powers; for when they are equall, their roots also are equall; and contrarily, the Powers are equall, when their roots are equal. But if the question be much complicated, there cannot by any of these wayes be constituted a certaine Rule, from the supposition of which of the unknown quan∣tities the Analysis may best begin; nor out of the variety of Equa∣tions that at first appeare, which we were best to choose; but the successe will depend upon dexterity, upon formerly ac∣quired Science, and many times upon fortune.

For no man can ever be a good Analyst without being first a good Geometrician; nor do the rules of Analysis make a Geo∣metrician, as Synthesis doth; which begins at the very Elements, and proceeds by a Logical Use of the same. For the true teaching of Geometry is by Synthesis, according to Euclides method; and he that hath Euclide for his Master, may be a Geometrician without Vieta (though Vieta was a most admirable Geometrician); but he that has Vieta for his master, not so, without Euclide.

And as for that part of Analysis which works by the Powers, though it be esteemed by some Geometricians (not the chiefest) to be the best way of solving all Problemes, yet it is a thing of no great extent; it being all contained in the doctrine of rectangles, and rectangled Solids. So that although they come to an Equati∣on which determines the quantity sought, yet they cannot some∣times by art exhibit that quantity in a Plain, but in some Conique Section; that is, as Geometricians say, not Geometrically, but me∣chanically. Now such Problemes as these, they call Solid; and when they cannot exhibit the quantity sought for with the helpe of a conique Section, they call it a Lineary Probleme. And therefore in the quantities of angles, and of the arches of Circles, there is no use at all of the Analyticks which proceed by the Powers; so that

the Antients pronounced it impossible, to exhibit in a plaine the Division of Angles, except bisection, and the bisection of the bisected parts, otherwise then mechanically. For Pappus, (before the 31 proposition of his fourth Book) distinguishing and defining the several kinds of Problemes, says that some are Plain, others Solid, and others Lineary. Those therefore which may be solved by straight lines and the circumferences of Circles (that is, which may be described with the Rule and Compass, without any other In∣strument) are fitly called Plain; for the lines by which such Problemes are found out, have their generation in a Plain. But those which are solved by the using of some one or more Conique Sections in their con∣struction, are called Solid, because their construction cannot be made with∣out using the superficies of solid figures, namely of Cones. There remains the third kinde, which is called Lineary, because other lines besides those already mentioned are made use of in their construction, &c. And a little after he sayes, Of this kinde are the Spiral lines, the Qua∣dratrices, the Conchoeides, and the Cissoeides. And Geometri∣cians think it no small fault, when for the finding out of a Plain Probleme any man makes use of Coniques, or new Lines. Now he ranks the Trisection of an angle among Solid Problemes, and the Quinquesection among Lineary. But what! are the ancient Geome∣tricians to be blamed, who made use of the Quadratrix for the finding out of a straight line equal to the arch of a Circle? and Pappus himself, was he faulty when he found out the trisection of an Angle by the help of an Hyperbole? Or am I in the wrong, who think I have found out the construction of both these Problemes by the Rule and Compass onely? Neither they, nor I. For the Anci∣ents made use of this Analysis which proceeds by the Powers; and with them it was a fault to do that by a more remote Power, which might be done by a neerer; as being an argument that they did not sufficiently understand the nature of the thing. The virtue of this kind of Analysis consists in the changing and turning and tos∣sing of Rectangles and Analogismes; and the skill of Analysts is meer Logick, by which they are able methodically to find out whatsoever lies hid either in the Subject or Predicate of the Con∣clusiō sought for. But this doth not properly belong to Algebra, or the Analyticks Specious, Symbolical or Cossick; which are, as I may say, the Brachygraphy of the Analyticks, and an art, neither of teaching nor

learning Geometry, but of registring with brevity and celerity, the inventions of Geometricians. For though it be easie to discourse by Symbols of very remote propositions; yet whether such discourse deserve to be thought very profitable, when it is made without any Ideas of the things themselves, I know not.

CHAP. XXI. Of Circular Motion.

• 1 In Simple Motion, every Straight Line taken in the Body moved, is so car∣ried, that it is always parallel to the places in which it formerly was.
• 2 If Circular Motion be made about a resting Center, and in that Circle there be an Epicyle, whose revolution is made the contrary way, in such manner, that in equal times it make equal angles, every Straight Line taken in that Epicycle will be so carried, that it will alwayes be parallel to the places in which it for∣merly was.
• 3 The properties of Simple Motion.
• 4 If a fluid Body be moved with simple Circular Motion, all the points taken in it will de∣scribe their Circles in times proportional to the distances from the Center.
• 5 Simple Motion dissipates Heterogeneous and congregates Homogeneous Bodies.
• 6 If a Circle made by a Movent moved with Simple Motion, be commensurable to another Circle made by a point which is carried about by the same Movent, all the points of both the Circles will at some time return to the same situation.
• 7 If a Sphere have Simple Motion, its Motion will more dissipate Heterogeneous Bodies by how much it is more re∣mote from the Poles.
• 8 If the Simple Circular Motion of a fluid Body, be hindered by a Body which is not fluid, the fluid Body will spread it self upon the Superficies of that Body.
• 9 Circular Motion about a fixed Center, casteth off by the Tangent such things as lie upon the Circumference and stick not to it.
• 10 Such things as are moved with Simple Circular Moti∣tion, beget Simple Circular Motion.
• 11 If that which is so moved have one side hard, and the other side fluid, its Motion will not be perfectly Cir∣cular.

1 I Have already defined Simple Motion to be that, in which the several points taken in a moved Body, do in several equal times describe several equal arch∣es. And therefore in Simple Circular Motion it is necessary that every Straight Line taken in the Moved Body be alwayes carried parallel to itself; which I thus demonstrate.

First, let A B (in the first figure) be any Straight Line taken in

any Solid Body; and let AD be any arch drawn upon any Center C and Radius CA. Let the point B be understood to describe to∣wards the same parts the arch BE, like and equall to the arch AD. Now in the same time in which the point A transmits the arch AD, the point B (which by reason of its simple motion is supposed to be carried with velocity equall to that of A) will transmit the arch BE; and at the end of the same time the whole AB will be in DE; and therefore AB and DE are equall. And seeing the arches AD and BE are like and equall, their subtend∣ing straight lines AD and BE will also be equall; and therefore the four sided figure ABDE will be a parallelogram. Where∣fore AB is carried parallel to it selfe. And the same may be pro∣ved by the same method, if any other straight line be taken in the same moved Body in which the straight line AB was taken. So that all straight lines taken in a Body moved with Simple Ci••cular Motion will be carried parallel to themselves.

• Coroll. 1 It is manifest that the same will also happen in any Body which hath Simple Motion, though not Circular. For all the points of any straight line whatsoever, will describe lines though not Circular, yet equall; so that though the crook∣ed lines AD and BE were not arches of Circles, but of Para∣bolas, Ellipses, or of any other figures; yet both they, and their Subtenses, and the straight lines which joyne them, would be e∣qual and parallel.
• Coroll. 2 It is also manifest, that the Radii of the equall circles AD and BE, or the Axis of a Sphere, will be so carried, as to be allwayes parallel to the places in which they formerly were. For the straight line BF drawn to the center of the arch BE be∣ing equall to the Radius AC, will also be equall to the straight line FE or CD; and the angle BFE will be equall to the angle ACD. Now the intersection of the straight lines CA and BE, being at G, the angle CGE (seeing BE and AD are parallel) will be equal to the angle DAC. But the angle EBF is equal to the same angle DAC; and therefore the angles CGE and EBF are also equal. Wherefore AC and BF are parallel; which was to be demonstrated.

2 Let there be a Circle given (in the second figure), whose

center is A, and Radius AB; and upon the center B and any Ra∣dius BC let the Epicycle CDE be described. Let the center B be understood to be carried about the center A, and the whole E∣picycle with it till it be coincident with the Circle FGH, whose center is I; and let BAI be any angle given. But in the time that the center B is moved to I, let the Epicycle CDE have a contra∣ry revolution upon its own center, namely from E by D to C ac∣cording to the same proportions; that is, in such manner, that in both the Circles, equal angles be made in equal times. I say EC the Axis of the Epicycle will be alwayes carried parallel to it self. Let the angle FIG be made equal to the angle BAI; IF and AB will therefore be parallel; and how much the Axis AG has departed from its former place AC (the measure of which pro∣gression is the angle CAG, or CBD which I suppose equal to it), so much in the same time has the Axis IG (the same with BC) departed from its own former situation. Wherefore, in what time BC comes to IG by the motion from B to I upon the center A, in the same time G will come to F by the contrary motion of the Epicycle; that is, it will be turned backwards to F, & IG will lie in IF. But the angles FIG and GAC are equal; and therefore AC, that is, BC), and IG, (that is the Axis, though in different places) will be parallel. Wherefore, the Axis of the Epicycle EDC will be carried alwayes parallel to it self; which was to be proved.

Coroll. From hence it is manifest, that those two annual Motions which Copernicus ascribes to the Earth, are reducible to this one Circular Simple Motion, by which all the points of the moved Bo∣dy are carried always with equal velocity, that is, in equal times they make equal revolutions uniformly.

This, as it is the most simple, so it is the most frequent of all Circular Motions; being the same which is used by all men when they turn any thing round with their arms, as they do in grinding or sifting. For all the points of the thing moved, describe lines which are like and equal to one another. So that if a man had a Ruler, in which many Pens points of equal length were fastned, he might with this one Motion write many lines at once.

3 Having shewed what Simple Motion is, I will here also set down some properties of the same.

First, when a Body is moved with Simple Motion in a fluid Me∣dium which hath no vacuity, it changes the situation of all the parts of the fluid ambient which resist its motion; I say there are no parts so small of the fluid ambient, how farre soever it be con∣tinued, but do change their situation, in such manner, as that they leave their places continually to other small parts that come into the same.

For (in the same second figure) let any Body, as KLMN, be understood to be moved with Simple Circular Motion; and let the Circle which every point thereof describes have any determined quantity, suppose that of the same KLMN. Wherefore the Cen∣ter A, and every other point, and consequently the moved Body it self, will be carried sometimes towards the side where is K, and sometimes towards the other side where is M. When therefore it is carried to K, the parts of the fluid Medium on that side will go back; and (supposing all space to be full) others on the other side will succeed. And so it will be when the Body is carried to the side M, and to N, and every way. Now when the neerest parts of the fluid Medium go back, it is necessary that the parts next to those neerest parts go back also; and (supposing still all space to be full) other parts will come into their places with succession per∣petual and infinite. Wherefore all, even the least parts of the fluid Medium change their places, &c. which was to be proved.

It is evident from hence, that Simple Motion, whether Circu∣lar, or not Circular, of Bodies which make perpetual returns to their former places, hath greater or less force to dissipate the parts of resisting Bodies, as it is more or less swift, and as the lines described have greater or less magnitude. Now the greatest Velo∣city that can be, may be understood to be in the least circuit, and the least in the greatest; and may be so supposed when there is need.

4 Secondly, supposing the same Simple Motion in the Aire, Water, or other fluid Medium; the parts of the Medium which adhere to the Moved Body will be carried about with the same Motion and Velocity, so that in what time soever any point of the Movent finishes its Circle, in the same time every part of the Medium which adheres to the Movent, shall also de∣scribe

such a part of its Circle, as is equal to the whole Circle of the Movent; I say it shall describe a part, and not the whole Circle, because all its parts receive their motion from an interiour concentrique Movent, and of Concentrique Circles the exteriour are alwayes greater then the interiour; nor can the mo∣tion imprinted by any Movent be of greater Velocity then that of the Movent it self. From whence it follows, that the more remote parts of the fluid ambient, shall finish their Circles in times which have to one another the same proportion with their distances from the Movent. For every point of the fluid ambient, as long as it toucheth the Body which carries it about, is carried about with it, and would make the same Circle, but that it is left behind so much as the exteriour Circle exceeds the interiour. So that if we suppose some thing which is not fluid to float in that part of the fluid ambient which is neerest to the Movent, it will together with the Movent be carried about. Now that part of the fluid am∣bient which is not the neerest but almost the neerest, receiving its degree of velocity from the neerest, (which degree cannot be greater then it was in the giver) doth therefore in the same time make a Circular Line, not a whole Circle, yet equal to the whole Circle of the neerest. Therefore in the same time that the Movent describes its Circle, that which doth not touch it shall not describe its Circle; yet it shall describe such a part of it, as is equal to the whole Circle of the Movent. And after the same manner, the more remote parts of the ambient will describe in the same time such parts of their Circles as shall be severally equal to the whole Cir∣cle of the Movent; and by consequent they shall finish their whole Circles in times proportional to their distances from the Movent; which was to be proved.

5 Thirdly, The same Simple Motion of a Body placed in a fluid Medium, congregates, or gathers into one place such things as na∣turally float in that Medium, if they be Homogeneous; and if they be Heterogeneous, it separates and dissipates them. But if such things as be Heterogeneous do not float, but settle, then the same Motion stirs and mingles them disorderly together. For seeing Bodies which are unlike to one another, that is, Heterogeneous Bodies, are not unlike in that they are Bodies (for Bodies, as Bodies,

have no difference) but onely from some special Cause, that is, from some internal Motion, or Motions of their smallest parts (for I have shewn in the 9th Chapter and 9th Article, that all Mutation is such Motion), it remains that Heterogeneous Bodies have their unlikeness or difference from one another from their internal or specifical Motions. Now Bodies w^ch have such difference, receive unlike & different Motions from the same external common Mo∣vent; and therefore they will not be moved together, that is to say, they will be dissipated. And being dissipated they will neces∣sarily at some time or other meet with Bodies like themselves, and be moved alike and together with them; and afterwards meeting with more Bodies like themselves, they will unite and become greater Bodies. Wherefore Homogeneous Bodies are congre∣gated, and Heterogenous dissipated by Simple Motion in a Me∣dium where they naturally float. Again, such as being in a fluid Medium, do not float, but sink, if the Motion of the fluid Medium be strong enough, will be stirred up and carried away by that Mo∣tion, and consequently they will be hindred from returning to that place to which they sink naturally, and in which onely they would unite, and out of which they are promiscuously carried; that is, they are disorderly mingled.

Now this Motion by which Homogeneous Bodies are congrega∣ted, and Heterogeneous are scattered, is that which is commonly called Fermentation, from the Latine Fervere; as the Greeks have their 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉 (which signifies the same) from 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉 Ferveo. For Seeth∣ing makes all the parts of the Water change their places; and the parts of any thing that is thrown into it, will go several wayes ac∣cording to their several natures. And yet all Fervor or Seething is not caused by Fire; for New Wine and many other things have also their Fermentation and Fervor, to which Fire contributes lit∣tle, and some times nothing. But when in Fermentation we find Heat, it is made by the Fermentation.

6 Fourthly, in what time soever the Movent whose Center is A (in the 2d figure) moved in KLN shall by any number of revo∣lutions (that is, when the Perimeters BI and KLN be commensu∣rable) have described a Line equal to the Circle which passes through the points B and I; in the same time all the points of the

floating Body whose Center is B, shall return to have the same si∣tuation in respect of the Movent, from which they departed. For seeing it is as the distance BA, that is, as the Radius of the Circle which passes through BI, is to the Perimeter it self BI, so the Ra∣dius of the Circle KLN is to the Perimeter KLN; and seeing the velocities of the points B and K are equal, the time also of the re∣volution in IB to the time of one revolution in KLN, will be as the Perimeter BI to the Perimeter KLN; and therefore so many revolutions in KLN as together taken are equal to the Perimeter BI, will be finished in the same time in which the whole Perime∣ter BI is finished; & therefore also the points L, N, F & H, or any of the rest, will in the same time return to the same situation from which they departed; and this may be demonstrated whatsoever be the points considered. Wherefore all the points shall in that time return to the same situation; which was to be proved.

From hence it follows, that if the Perimeters BI and LKN be not commensurable, then all the points wil never return to have the same situation or configuration in respect of one another.

7 In Simple Motion, if the Body moved be of a Spherical fi∣gure, it hath less force towards its Poles then towards its middle, to dissipate Heterogeneous, or to congregate Homogeneous Bo∣dies.

Let there be a Sphere (as in the third figure) whose Center is A and Diameter BC; & let it be conceived to be moved with Simple Circular Motion; of which Motion let the Axis be the Straight Line DE, cutting the Diameter BC at right angles in A. Let now the Circle which is described by any point B of the Sphere, have BF for its Diameter; and taking FG equal to BC, and dividing it in the middle in H, the Center of the Sphere A, will when half a revolution is finished, lie in H. And seeing HF and AB are e∣qual, a Circle described upon the Center H with the Radius HF or HG, will be equal to the Circle whose Center is A and Radius AB. And if the same Motion be continued, the point B will at the end of another half revolution return to the place from whence it began to be moved; and therefore at the end of half a revoluti∣on, the point B will be carried to F, and the whole Hemisphere DBE into that Hemisphere in which are the points L, K and F. Wherfore that part of the fluid Medium which is cōtiguous to the

point F, will in the same time go back the length of the Straight Line BF; and in the return of the point F to B, that is, of G to C, the fluid Medium wil go back as much in a Straight Line from the point C. And this is the effect of Simple Motion in the middle of the Sphere, where the distance from the Poles is greatest. Let now the point I be taken in the same Sphere neerer to the Pole E, and through it let the Straight Line IK be drawn parallel to the Straight Line BF, cutting the arch FL in K, & the Axis HL in M; then connecting HK, upon HF let the perpendicular KN be drawn. In the same time therefore that B comes to F, the point I will come to K, BF and IK being equal, and described with the same velo∣city. Now the Motion in IK to the fluid Medium upon which it works, namely to that part of the Medium which is contiguous to the point K, is oblique, whereas if it proceeded in the Straight Line HK, it would be perpendicular; and therefore the Motion which proceeds in IK has less power, then that which proceeds in HK with the same velocity. But the Motions in HK and HF do equally thrust back the Medium; and therefore the part of the Sphere at K, moves the Medium less, then the part at F; namely so much less, as KN is less then HF. Wherefore also the same Motion hath less power to disperse Heterogeneous, and to congre∣gate Homogeneous Bodies, when it is neerer, then when it is more remote from the Poles; which was to be proved.

Corollary. It is also necessary, that in Plains which are perpendi∣cular to the Axis, and more remote then the Pole it self from the middle of the Sphere, this Simple Motion have no effect. For the Axis DE with Simple Motion describes the Superficies of a Cy∣linder; and towards the Bases of the Cylinder there is in this Motion no endeavour at all.

8 If in a fluid Medium, moved about (as hath been said) with Simple Motion, there be conceived to float some other Spherical Body which is not fluid, the parts of the Medium which are stop∣ped by that Body, will endeavour to spread themselves every way upon the Superficies of it. And this is manifest enough by expe∣rience, namely by the spreading of water poured out upon a pave∣ment. But the reason of it may be this. Seeing the Sphere A (in the 3d figure) is moved towards B, the Medium also in which it is moved, will have the same Motion. But because in this Motion

it falls upon a Body not liquid, as G, so that it cannot go on; and seeing the small parts of the Medium can not go forwards, nor can they go directly backwards; against the force of the Movent; it re∣mayns therefore that they diffuse themselves upon the Superficies of that Body, as towards O and P, Which was to be proved.

9 Compounded Circular Motion (in which all the parts of the moved Body do at once describe Circumferences, some great∣er, others less, according to the proportion of their several distances from the common Center) carries about with it such Bodies, as being not fluid, adhere to the Body so moved; and such as do not adhere, it casteth forwards in a Straight Line which is a Tangent to the point from which they are cast off.

For let there be a Circle whose Radius is AB (in the fourth fi∣gure); and let a Body be placed in the Circumference in B, which if it be fixed there, will necessarily be carried about with it, as is manifest of it self. But whilest the motion proceeds, let us suppose that Body to be unfixed in B. I say the Body wil cōtinue its motion in the Tangent BC. For let both the Radius AB, and the Sphere B, be conceived to consist of hard matter; and let us suppose the Ra∣dius AB to be stricken in the point B by some other Body which falls upon it in the Tangent DB. Now therefore there will be a motion made by the concourse of two things, the one, Endeavour towards C in the Straight Line DB produced, (in which the Bo∣dy B would proceed, if it were not retained by the Radius AB); the other, the Retention it self. But the Retention alone causeth no endeavour towards the Center; and therefore the Retention being taken away, (which is done by the unfixing of B) there will remain but one Endeavour in B, namely, that in the Tangent BC. Wherefore the Motion of the Body B unfixed, will proceed in the Tangent BC; which was to be proved.

By this demonstration it is manifest, that Circular Motion a∣bout an unmoved Axis, shakes off, and puts further from the Cen∣ter of its motion such things as touch, but do not stick fast to its Superficies; and the more, by how much the distance is greater from the Poles of the Circular Motion; and so much the more al∣so, by how much the things that are shaken off, are less driven to∣wards the Center by the fluid ambient, for other Causes.

10 If in a fluid Medium a Spherical Body be moved with sim∣ple Circular Motion; and in the same Medium there float another Sphere whose matter is not fluid; this Sphere also shall be moved with simple Circular Motion.

Let BCD (in the 5th figure) be a Circle, whose Center is A, and in whose Circumference there is a Sphere so moved that it describes with Simple Motion the Perimeter BCD. Let also EFG be another Sphere of Consistent matter, whose Semidia∣meter is EH, and Center H; and with the Radius AH let the Circle HI be described. I say the Sphere EFG will (by the Motion of the Body in BCD) be moved in the Circumference HI with Simple Motion.

For seeing the Motion in BCD (by the 4th Article of this Chapter) makes all the points of the fluid Medium describe in the same time Circular Lines equal to one another, the points E, H and G of the Straight Line EHG will in the same time describe with equal Radii equal Circles. Let EB be drawn equal and pa∣rallel to the Straight Line AH; and let AB be connected, which will therefore be equal and parallel to EH; and therefore also, if upon the Center B and Radius BE the arch EK be drawn equal to the arch HI, and the straight Lines AI, BK and IK be drawn, BK and AI will be equal; and they will also be parallel, because the two arches EK and HI, that is, the two angles KBE and IAH are equal; and consequently the Straight Lines AB and KI which connect them will also be equal and parallel. Wherefore KI and EH are parallel. Seeing therefore E and H are carried in the same time to K and I, the whole Straight Line IK will be parallel to EH, from whence it departed. And therefore, (seeing the Sphere EFG is supposed to be of consistent matter, so as all its points keep alwayes the same situation) it is necessary that every other Straight Line taken in the same Sphere, be carried alwayes paral∣lel to the places in which it formerly was. Wherefore the Sphere EFG is moved with simple Circular Motion; which was to be demonstrated.

11 If in a fluid Medium, whose parts are stirred by a Body mo∣ved with Simple Motion, there float annother Body, which hath its Superficies either wholly hard, or wholly fluid; the parts of

this Body shall approach the Center equally on all sides, that is to say, the motion of the Body shall be Circular, and Concentrique with the motion of the Movent. But if it have one side hard, and the other side fluid, then both those Motions shall not have the same center, nor shall the floating Body be moved in the Circum∣ference of a perfect Circle.

Let a Body be moved in the Circumference of the Circle KL MN (in the 2d figure) whose center is A. And let there be another Body at I, whose Superficies is either all hard, or all fluid. Also let the Medium in which both the Bodies are placed, be fluid. I say the Body at I will be moved in the Circle IB about the Center A. For this has been demonstrated in the last Article.

Wherefore let the Superficies of the Body at I, be fluid on one side, and hard on the other. And first, let the fluid side be towards the Center. Seeing therefore the Motion of the Medium is such, as that its parts do continually change their places, (as hath been shewn in the 5th Article); if this change of place be considered in those parts of the Medium which are contiguous to the fluid Su∣perficies, it must needs be, that the small parts of that Super∣ficies enter into the places of the small parts of the Medium which are contiguous to them; And the like change of place will be made with the next contiguous parts towards A. And if the fluid parts of the Body at I, have any degree at all of tenacity (for there are degrees of tenacity, as in the Aire and Water) the whole fluid side will be lifted up a little; but so much the less, as its parts have less tenacity; whereas the hard part of the Superficies which is contiguous to the fluid part, has no cause at all of elevation, that is to say, no endeavour towards A.

Secondly, let the hard Superficies of the Body at I, be towards A. By reason therefore of the said change of place of the parts which are contiguous to it, the hard Superficies must of necessi∣ty (seeing by Supposition there is no empty Space) either come neerer to A, or else its smallest parts must supply the contiguous places of the Medium, which otherwise would be empty. But this cannot be by reason of the supposed hardness; and therefore the other must needs be, namely, that the Body come neerer to A. Wherefore the Body at I, has greater endeavour towards the cen∣ter

A, when its hard side is next it, then when it is averted from it. But the Body in I, while it is moving in the circumference of the Circle IB, has sometimes one side, sometimes another turned to∣wards the center; and therefore it is sometimes neerer, sometimes further off from the center A. Wherefore the Body at I, is not car∣ried in the circumference of a perfect Circle; which was to be de∣monstrated.

CHAP. XXII. Of other Variety of Motion.

• 1 Endeavour and Pressure how they differ.
• 2 Two kinds of Mediums in which Bodies are moved.
• 3 Propagation of Motion what it is
• 4 What motion Bodies have when they press one another.
• 5 Fluid Bodies, when they are pressed together, penetrate one another.
• 6 When one Body presseth an∣other, and doth not penetrate it, the action of the pressing Body is perpendi∣cular to the Superficies of the Body pressed.
• 7 When a hard Body, pressing another Body, penetrates the same, it doth not penetrate it perpendicularly, unless it fall perpendicularly upon it.
• 8 Motion sometimes opposite to that of the Movent.
• 9 In a full Medium, Motion is propagated to any di∣stance.
• 10 Dilatation and Contraction what they are.
• 11 Dilatation and Contraction suppose Mutation of the smallest parts in respect of their si∣tuation.
• 12 All Traction is Pulsion.
• 13 Such things as being pres∣sed, or bent, restore themselves, have motion in their internal parts.
• 14 Though that which carrieth another be stopped, the Body carried will proceed.
• 15, 16 The effects of Percussion not to be compared with those of Waight.
• 17, 18 Motion cannot begin first in the internal parts of a Bo∣dy.
• 19 Action and Reaction proceed in the same Line.
• 20 Habit what it is.

1 I Have already (in the 15th Chap. at the 2d Article) defined Endeavour to be Motion through some Length, though not considered as Length, but as a Point. Whether therefore there be resistance or no resistance, the Endeavour will still be the same. For simply to Endeavour, is to Go. But when two Bodies having op∣posite Endeavours press one another, then the Endeavour of ei∣ther of them is that which we call Pressure, and is mutual when their pressures are opposite.

2. Bodies moved, and also the Mediums in which they are mo∣ved, are of two kinds. For either they have their parts coherent in such manner, as no part of the Moved Body will easily yeild to the Mouent, except the whole Body yeild also, and such are the things

we call Hard; Or else their parts, while the whole remains un∣moved, will easily yeild to the Movent; and these we call Fluid or Soft Bodies. For the words Fluid, Soft, Tough and Hard (in the same manner as Great and Little) are used onely comparatively; and are not different kinds, but different degrees of Quality.

3 To Do, and to Suffer is to Move and to be moved; and nothing is moved, but by that which toucheth it, and is also moved, (as has been formerly shewn). And how great sover the distance be, we say the first Movent moveth the last moved Body; but mediate∣ly; namely so, as that the first moveth the second, the second the third, and so on, till the last of all be touched. When therefore one Body having opposite Endeavour to another Body, moveth the same, and that moveth a third, and so on, I call that action Propa∣gation of Motion.

4 When two fluid Bodies which are in a free and open Space, press one another, their parts will endeavour, or be moved to∣wards the sides, not onely those parts which are there where the mutual contact is, but all the other parts. For in the first contact, the parts which are pressed by both the endeavouring Bodies, have no place either forwards or backwards in which they can be mo∣ved; and therefore they are pressed out towards the sides. And this expressure, when the forces are equal, is in a line perpendicular to the Bodies pressing. But whensoever the formost parts of both the Bodies are pressed, the hindermost also must be pressed at the same time; for the motion of the hindermost parts cannot in an instant be stopped by the resistance of the formost parts, but pro∣ceeds for some time; and therefore seeing they must have some place in which they may be moved, and that there is no place at all for them forwards, it is necessary that they be moved into the places which are towards the sides every way. And this effect followes of necessity, not onely in Fluid, but in Consistent and Hard Bodies, though it be not alwayes manifest to sense. For though from the compression of two stones we cannot with our eyes discerne any swelling outwards towards the sides, (as we perceive in two Bodies of wax;) yet we know well enough by reason, that some tumor must needs be there, though it be but little.

5 But when the Space is enclosed, and both the Bodies be fluid, they will (if they be pressed together) penetrate one ano∣teer, though differently according to their different endeavours. For suppose a hollow Cylinder of hard matter, well stopped at both ends, but filled first, below with some heavy fluid Body, as Quicksilver; and above with Water or Aire. If now the bot∣tome of the Cylinder be turned upwards, the heaviest fluid Body which is now at the top, having the greatest endeavour down∣wards, and being by the hard sides of the vessel hindered from extending it selfe sidewayes, must of necessity either be received by the lighter Body, that it may sink through it, or else it must open a passage through it selfe, by which the lighter Body may ascend. For of the two Bodies, that whose parts are most easily se∣parated, will the first be divided; which being done, it is not ne∣cessary that the parts of the other, suffer any separation at all. And therefore when two Liquours which are enclosed in the same ves∣sel, change their places, there is no need that their smallest parts should be mingled with one another; for a way being opened through one of them, the parts of the other need not be separa∣ted.

Now if a fluid Body which is not enclosed press a hard Body, its endeavour will indeed be towards the internal parts of that hard Body; but (being excluded by the resistance of it) the parts of the fluid Body will be moved every way according to the Superficies of the hard Body, and that equally, if the pressure be perpendicu∣lar; for when all the parts of the Cause are equal, the Effects will be equal also. But if the pressure be not perpendicular, then the angles of Incidence being unequal, the expansion also will be un∣equal, namely, greater on that side where the angle is greater, be∣cause that motion is most direct which proceeds by the directest Line.

6 If a Body, pressing another Body do not penetrate it, it will ne∣vertheless give to the part it presseth, an endeavour to yeild and recede in a straight line perpendicular to its Superficies in that point in which it is pressed.

Let ABCD (in the first figure) be a hard Body; and let ano∣ther Body, falling upon it in the straight line EA, with any inclina∣tion, or without inclination, press it in the point A. I say the Body so

pressing, & not penetrating it, will give to the part A an endeavour to yeild or recede in a straight Line perpendicular to the line AD.

For let AB be perpendicular to AD; and let BA be produced to F. If therefore AF be coincident with AE, it is of it self mani∣fest that the motion in EA will make A to endeavour in the line AB. Let now EA be oblique to AD; and from the point E let the straight line EC be drawn, cutting AD at right angles in D; and let the rectangles ABCD and ADEF be completed. I have shewn (in the 8th Article of the 16th Chapter) that the Bo∣dy will be carried from E to A by the concourse of two Uniform Motions, the one in EF and its parallels, the other in ED and its parallels. But the motion in EF and its parallels (whereof DA is one) contributes nothing to the Body in A, to make it endeavour or press towards B; and therefore the whole endeavour which the Body hath in the inclined line EA, to pass, or press the Straight line AD, it hath it all from the perpendicular motion or endeavour in FA. Wherefore the Body E after it is in A, will have onely that perpendicular endeavour which proceeds from the motion in FA, that is, in AB; which was to be proved.

7 If a hard Body falling upon, or pressing another Body, pene∣trate the same, its endeavour after its first penetration will be nei∣ther in the inclined line produced, nor in the perpendicular, but sometimes betwixt both, sometimes without them.

Let EAG (in the same •• figure) be the inclined line produced; and First, let the passage through the Medium in which EA is, be easier then the passage through the Medium in which AG is. As soon therefore as the Body is within the Medium in which is AG, it will finde greater resistance to its motion in DA and its parallels, then it did whilest it was above AD; and therefore be∣low AD it will proceed with slower motion in the parallels of DA, then above it. Wherefore the motion which is compounded of the two motions in EF and ED will be slower below AD, then above it; and therefore also, the Body will not proceed from A in EA produced, but below it. Seeing therefore the endeavour in AB is generated by the endeavour in FA; if to the endeavour in FA there be added the endeavour in DA, (which is not all taken away by the immersion of the point A into the lower Medium) the Body will not proceed from A in the perpendicular AB, but be∣yond

it, namely, in some straight line between AB and AG, as in the line AH.

Secondly, let the passage through the Medium EA, be less easie then that through AG. The motion therefore which is made by the concourse of the motions in EF and FB, is slower above AD then below it; and consequently, the endeavour will not proceed from A in EA produced, but beyond it, as in AI. Wherefore, If a hard Body falling, &c; which was to be proved.

This Divergency of the Straight line AH from the straight line AG, is that which the Writers of Opticks commonly call Refraction; which, when the passage is ea••ier in the first then in the second Me∣dium, is made by diverging from the line of Inclination towards the perpendicular; and contrarily, when the passage is not so easie in the first Medium, by departing farther from the perpendicular.

8 By the 6th Theoreme it is manifest, that the force of the Mo∣vent may be so placed, as that the Body moved by it, may proceed in a way almost directly contrary to that of the Movent; as we see in the motion of Ships.

For let AB (in the 2d figure) represent a Ship, whose length from the prow to the poop is AB; and let the winde lie upon it in the straight parallel lines CB, DE and FG; and let DE and FG be cut in E and G by a straight Line drawn from B perpendicular to AB; also let BE and EG be equal, and the angle ABC any an∣gle how small soever. Then between BC and BA let the straight line BI be drawn; and let the Sail be conceived to be spred in the same line BI, and the winde to fall upon it in the points L, M and B; from which points, perpendicular to BI, let BK, MQ and LP be drawn. Lastly, let EN and GO be drawn perpendicular to BG, and cutting BK in H and K; and let HN and KO be made equal to one another, and severally equal to BA. I say the Ship BA by the winde falling upon it in CB, DE, FG, and other lines parallel to them, will be carried forwards almost opposite to the winde, that is to say, in a way almost contrary to the way of the Movent.

For the Winde that blowes in the Line CB, will (as hath been shewn in the 6th Article) give to the point B an endeavour to pro∣ceed in a straight line perpendicular to the straight line BI, that

is, in the straight line BK; and to the points M and L an endea∣vour to proceed in the straight lines MQ and LP, which are pa∣rallel to BK. Let now the measure of the time be BG, which is divided in the middle in E; & let the point B be carried to H in the time BE. In the same time therefore by the wind blowing in DM & FL (and as many other lines as may be drawn parallel to them) the whole Ship will be applyed to the straight line HN. Also at the end of the second time EG, it will be applyed to the straight line KO. Wherefore the Ship will always go forwards; and the angle it makes with the winde will be equal to the angle ABC, how small soever that angle be; and the way it makes will in eve∣ry time be equal to the straight line EH. I say thus it would be, if the Ship might be moved with as great celerity sidewayes from BA towards KO, as it may be moved forwards in the line BA. But this is impossible, by reason of the resistance made by the great quantity of water which presseth the side, much exceeding the re∣sistance made by the much smaller quantity which presseth the prow of the Ship; so that the way the Ship makes sidewayes is scarce sensible; and therefore the point B will proceed almost in the very line BA, making with the winde the angle ABC, how acute soever, that is to say, it will proceed almost in the straight line BC, that is, in a way almost contrary to the way of the Mo∣vent; which was to be demonstrated.

But the Sayl in BI must be so stretched, as that there be left in it no bo••ome at all; for otherwise the straight lines LP, MQ & BK will not be perpendicular to the plain of the Sayl, but falling be∣low P, Q and K will drive the Ship backwards. But by making use of a small Board for a Sayl, a little Waggon with wheels for the Ship, and of a smooth Pavement for the Sea, I have by experience found this to be so true, that I could scarce oppose the board to the winde in any obliquity though never so small, but the Waggon was carried forwards by it.

By the same 6th. Theoreme, it may be found, how much a stroke which falls obliquely, is weaker then a stroke falling perpendicu∣larly, they being like and equal in all other respects.

Let a stroke fall upon the Wall AB obliquely, as (for example) in the straight line CA (in the 3d figure). Let CE be drawn pa∣rallel

to AB, & DA perpendicular to the same AB & equal to CA; & let both the velocity & time of the motion in CA be equal to the velocity & time of the motion in DA. I say the stroke in CA will be weaker then that in DA in the proportion of EA to DA. For pro∣ducing DA howsoever to F, the endeavour of both the strokes will (by the 6th Art.) proceed from A in the perpendicular AF. But the stroke in CA is made by the concourse of two motions in CE and EA; of which that in CE contributes nothing to the stroke in A, because CE and BA are parallels; and therefore the stroke in CA is made by the motion which is in EA onely. But the velo∣city or force of the perpendicular stroke in EA, to the velocity or force of the stroke in DA, is as EA to DA. Wherefore the ob∣lique stroke in CA is weaker then the perpendicular stroke in DA, in the proportion of EA to DA or CA; Which was to be proved.

9 In a full Medium, all Endeavour proceeds as far as the Me∣dium it self reacheth; that is to say, if the Medium be infinite, the Endeavour will proceed infinitely.

For whatsoever Endeavoureth, is Moved, and therefore what∣soever standeth in its way, it maketh it yeild, at least a little, name∣ly so far as the Movent it self is moved forwards. But that which yeildeth is also moved, and consequently maketh that to yeild which is in its way, and so on successively as long as the Medium is full; that is to say, infinitely, if the full Medium be infinite, which was to be proved.

Now although Endeavour thus perpetually propagated, do not alwayes appear to the Senses as Motion; yet it appears as Action, or as the efficient cause of some Mutation. For if there be placed be∣fore our Eyes some very little object, as (for example) a small grain of sand, which at a certain distance is visible; it is manifest that it may be removed to such a distance as not to be any longer seen, though by its action it still work upon the organs of sight, as is ma∣nifest from that (which was last proved) that all Endeavour pro∣ceeds infinitely. Let it be conceived therefore to be removed from our Eyes to any distance how great soever, and a sufficient number of other grains of sand of the same bigness added to it; it is evi∣dent that the aggregate of all those sands will be visible; and though none of them can be seen when it is single and severed

from the rest, yet the whole heap or hill which they make wil ma∣nifestly appear to the sight; which would be impossible if some a∣ction did not proceed from each several part of the whole heap.

10 Between the degrees of Hard and Soft, are those things which we call Tough, Tough being that, which may be bended with∣out being altered from what it was; and the Bending of a Line, is either the adduction or diduction of the extreme parts, that is, a morion from Straightness to Crookedness, or contrarily, whilest the line remains still the same it was; for by drawing out the ex∣treme points of a line to their greatest distance, the line is made straight, which otherwise is Crooked. So also the Bending of a Superficies, is the diduction or adduction of its extreme lines, that is, their Dilatation and Contraction.

11 Dilatation and Contraction, as also all Flexion supposes necessa∣rily that the internal parts of the Body bowed do either come nee∣rer to the external parts, or go further from them. For though Flexion be considered onely in the length of a Body, yet when that Body is bowed, the line which is made on one side will be convex, and the line on the other side will be concave; of which the con∣cave being the interiour line, will (unless something be taken from it and added to the convex line) be the more crooked, that is, the greater of the two. But they are equal; and therefore in Flexion there is an accession made from the interiour to the exteriour parts; and on the contrary, in Tension, from the exteriour to the interiour parts. And as for those things which do not easily suffer such transposition of their parts, they are called Brittle; and the great force they require to make them yield, makes them also with sudden motion to leap asunder, and break in pieces.

12 Also Motion is distinguished into Pulsion and Traction. And Pulsion, as I have already defined it, is when that which is moved, goes before that which moveth it. But contrarily, in Traction the Movent goes before that which is moved. Nevertheless, conside∣ring it with greater attention, it seemeth to be the same with Pul∣sion. For of two parts of a hard Body, when that which is foremost drives before it the Medium in which the motion is made, at the same time that which is thrust forwards, thrusteth the next, and this again the next, and so on successively. In which action, if we

suppose that there is no place void, it must needs be, that by con∣tinual Pulsion, namely, when that action has gone round, the Mo∣vent will be behind that part which at the first seemed not to be thrust forwards, but to be drawn; so that now the Body which was drawn, goes before the Body which gives it motion; and its motion is no longer Traction, but Pulsion.

13 Such things as are removed from their places by forcible Compression or Extension, and as soon as the force is taken a∣way, doe presently return and restore themselves to their for∣mer situation, have the beginning of their restitution within themselves, namely, a certain motion in their internal parts, which was there, when, before the taking away of the force, they were compressed, or extended. For that Restitution is mo∣tion, and that which is at rest cannot be moved, but by a mo∣ved and a Contiguous Movent. Nor doth the cause of their Resti∣tution proceed from the taking away of the force by which they were compressed or extended; for the removing of impediments hath not the efficacy of a cause (as has been shewn at the end of the 3d Article of the 15th Chapter). The Cause therefore of their Restitution, is some motion either of the parts of the Ambient; or of the parts of the Body compressed or extended. But the parts of the Ambient have no endeavour which contributes to their Com∣pression or Extension, nor to the setting of them at liberty, or Re∣stitution. It remayns therefore that from the time of their Com∣pression or Extension there be left some endeavour (or motion) by which, the impediment being removed, every part resumes its for∣mer place; that is to say, the whole Restores it self.

14 In the Carriage of Bodies if that Body which carries ano∣ther, hit upon any obstacle, or be by any means suddenly stopped, and that which is carried be not stopped, it will go on, till its mo∣tion be by some external impediment taken away.

For I have demonstrated in the 8th Chapter at the 19th Arti∣cle, that Motion, unless it be hindred by some external resistance, will be continued eternally with the same celerity; and in the 7th Article of the 9th Chap. that the action of an external Agent is of no effect without contact. When therefore that which carrieth another thing, is stopped, that stop doth not presently take away the motion of that which is carried. It will therefore proceed,

till its motion be by little and little extinguished by some external resistance; Which was to be proved; Though experience alone had been sufficient to prove this.

In like manner, if that Body which carrieth another be put from rest into sudden motion; that which is carried will not be moved forwards together with it, but will be left behind. For the contiguous part of the Body carried, hath almost the same motion with the Body which carries it; and the remote parts will receive different Velocities according to their different distances from the Body that carries them; namely, the more remote the parts are, the less will be their degrees of Velocity. It is necessary therefore that the Body which is carried, be left accordingly more or less behind. And this also is manifest by experience, when at the start∣ing forward of the Horse, the Rider falleth backwards.

15 In Percussion therefore, when one hard Body is in some small ••art of it stricken by another with great force, it is not necessary that the whole Body should yeild to the stroke with the same ce∣lerity with which the stricken part yeilds. For the rest of the parts receive their motion from the motion of the part stricken and yeilding, which motion is less propagated every way towards the sides then it is directly forwards. And hence it is, that sometimes very hard Bodies, which being erected can hardly be made to stand, are more easily broken, then thrown down by a violent stroke; when nevertheless, if all their parts together were by any weak motion thrust forwards they would easily be cast down.

16 Though the difference between Trusion and Percussion con∣sist onely in this, that in Trusion the motion both of the Movent and Moved Body begin both together in their very contact; and in Percussion the striking Body is first moved, and afterwards the Body stricken; Yet their Effects are so different, that it seems scarce possible to compare their forces with one another. I say, any ef∣fect of Percussion being propounded, as for example the stroke of a Beetle of any weight assigned, by which a Pile of any given length, is to be driven into earth of any tenacity given, it seems to me very hard if not impossible to define, with what weight, or with what stroke, and in what time, the same pile may be driven 〈◊〉〈◊〉 a depth assigned into the same earth. The cause of which dif∣ficulty

is this, that the velocity of the Percutient is to be compa∣red with the magnitude of the Ponderant. Now Velocity, seeing it is computed by the length of space transmitted, is to be accoun∣ted but as one Dimension; but Waight, is as a solid thing, being measured by the dimension of the whole Body. And there is no comparison to be made of a Solid Body with a Length, that is, with a Line.

17 If the internal parts of a Body be at rest, or retain the same situation with one another for any time how little soever, there cannot in those parts be generated any new motion, or endeavour, whereof the efficient cause is not without the Body of which they are parts. For if any small part which is comprehended within the Superficies of the whole Body, be supposed to be now at rest, and by and by to be moved, that part must of necessity receive its motion from some moved and contiguous Body. But (by sup∣position) there is no such moved and contiguous part within the Body. Wherefore, if there be any Endeavour or Motion, or change of situation, in the internal parts of that Body, it must needs arise from some efficient cause that is without the Body which contains them; Which was to be proved.

18 In hard Bodies therefore which are compressed or extend∣ed, if that which compresseth or extendeth them being taken a∣way, they restore themselves to their former place or situation, it must needs be, that that Endeavour (or Motion) of their internal parts, by which they were able to recover their former places or situations, was not extinguished when the force by which they were compressed or extended was taken away. Therefore when the Lath of a Cross-bow bent, doth, as soon as it is at liberty, re∣store it self, though to him that judges by Sense, both it and all its parts seem to be at rest; yet he that judging by Reason, doth not ac∣count the taking away of impediment for an efficient cause, nor conceives that without an efficient cause any thing can pass from Rest to Motion, will conclude, that the parts were already in moti∣on before they began to restore themselves.

19 Action and Reaction proceed in the same Line, but from op∣posite Terms. For seeing Reaction is nothing but Endeavour in the Patient to restore it self to that situation from which it was for∣ced

by the Agent; the endeavour or motion both of the Agent and Patient (or Reagent) will be propagated between the same terms, (yet so, as that in Action the Term from which, is in Reaction the Term to which). And seeing all Action proceeds in this manner, not onely between the opposite Terms of the whole line in which it is propagated, but also in all the parts of that line, the Terms from which and to which, both of the Action and Reaction, will be in the same line. Wherefore Action and Reaction proceed in the same line, &c.

20 To what has been said of Motion, I will add what I have to say concerning Habit. Habit therefore is a generation of Mo∣tion, not of Motion simply, but an easie conducting of the mo∣ved Body in a certain and designed way. And seeing it is attained by the weakning of such endeavours as divert its motion, there∣fore such endeavours are to be weakned by little and little. But this cannot be done but by the long continuance of action, or by actions often repeated; and therefore Custome begets that Faci∣cility, which is commonly and rightly called Habit; and it may be defined thus; HABIT is Motion made more easie and ready by Custome; that is to say, by perpetual endeavour, or by iterated endevours in a way differing from that in which the Motion proceeded from the begin∣ning, and opposing such endeavours as resist. And to make this more perspicuous by example, We may observe, that when one that has no skill in Musique, first, puts his hand to an Instrument, he cannot after the first stroke carry to his hand to the place where he would make the second stroke, without taking it back by a new endeavour, and as it were beginning again, pass from the first to the second. Nor will he be able to go on to the third place without another new endeavour, but he will be forced to draw back his hand again, and so successively, by renewing his en∣deavour at every stroke, till at the last by doing this often, and by compounding many interrupted motions or endeavours into one equal endeavour, he be able to make his hand go readily on from stroke to stroke in that order and way which was at the first designed. Nor are Habits to be observed in living creatures only, but also in Bodies inanimate. For we find, that when the Lath of a Crossbow is strongly bent, and would if the impedi∣ment

were removed return again with great force, if it re∣main a long time bent, it will get such a Habit, that when it is loosed and left to its own freedome, it will not onely not restore it self, but will require as much force for the bring∣ing of it back to its first posture, as it did for the bending of it at the first.

CHAP. XXIII. Of the Center of Equiponderation of Bodies pressing do••••ards in straight Parallel Lines.

• 1 Definitions and Suppositions.
• 2 Two Plains of Equiponderation are n•••• parallel.
• 3 The Center of Equiponderation is in every Plain of E∣quiponderation.
• 4 The Moments of equal Ponderants are to one another as their distances from the center of the Scale.
• 5,6. The Moments of unequal Ponderants have their proportion to one another compounded of the proportions of their Waights and distances from the center of the Scale re∣ciprocally taken.
• 7. If two Ponderants have their Moments and Di∣stances from the Center of the Scale in reciprocal proportion; they are e∣qually poised; and contrarily.
• 8 If the parts of any Ponderant press the Beam of the Scale every where equally, all the parts cut out off reckoned from the Center of the Scale▪ will have their Moments in the same propor∣tion with that of the parts of a Triangle cut off from the Vertex by straight Lines parallel to the base.
• 9 The Diameter of Equiponderation of Fi∣gures which are deficient▪ according to commensurable proportions of their altitudes and bases, divides the Axis, so, that the part taken next the vertex is to the other part as the complete figure to the deficient figure.
• 10 The diameter of Equiponderation of the Complement of the half of any of the said deficient figures, divides that line which is drawn▪ through the vertex parallel to the base, so, that the part next the vertex is to the other part as the complete figure to the Complement.
• 11 The Center of Equiponderati∣on of the half of any of the desicient figures in the first row of the Table of the 3d. Article of the 17th Chapter, may be found out by the numbers of the second row.
• 12 The center of Equiponderation of the half of any of the figures in the second row of the same Table, may be found out by the num∣bers of the fourth row.
• 13 The Center of Equiponderation of the half of any of the figures in the same Table, being known, the Center of the Ex∣cess of the same figure above a Triangle of the same altitude and base is also known.
• 14 The Center of Equiponderation of a solid Sector, is in

• the Axis, so divided, that the part next the Vertex be to the whole Axis want half the Axis of the portion of the Sphere, as 3 to 4.

1

Definitions.

• 1 A Scale, is a straight line, whose middle point is immoveable, all the rest of its points being at liberty; and that part of the Scale which reaches from the center to either of the waights, is called the Beam.
• 2 Equiponderation is, when the endeavour of one Body which pres∣ses one of the Beams, resists the endeavour of another Body pressing the other Beam, so, that neither of them is moved; and the Bodies when neither of them is moved, are said to be Equally poised.
• 3 Waight, is the aggregate of all the Endeavours, by which all the points of that Body which presses the Beam, tend downwards in lines parallel to one another; and the Body which presses, is called the Ponderant.
• 4 Moment, is the Power which the Ponderant has to move the Beam, by reason of a determined situation.
• 5 The plain of Equiponderation, is that, by which the Ponderant is so divided, that the Moments on both sides remain equal.
• 6 The Diameter of Equiponderation, is the common Section of the two Plains of Equiponderation; and is in the straight line by which the waight is hanged.
• 7 The Center of Equiponderation, is the common point of the two Diameters of Equiponderation.

Suppositions

• 1 When two Bodies are equally pois'd, if waight be added to one of them, and not to the other, their Equiponderation ceases.
• 2 When two Ponderants of equal magnitude, and of the same Species or matter, press the Beam on both sides at equal di∣stances from the center of the Scale, their Moments are e∣qual. Also when two Bodies endeavour at equal distances from the center of the Scale, if they be of equal magnitude and of the same Species, their Moments are equal.

2 No two Plains of Equiponderation are parallel.

Let A B C D (in the first figure) be any Ponderant whatsoever; and in it let E F be a Plain of Equiponderation; parallel to which, let any other Plain be drawn, as G H. I say G H is not a Plain of Equiponderation. For seeing the parts A E F D and E B C F of the Ponderant A B C D, are equally pois'd; and the weight E G H F is added to the part A E F D, and nothing is added to the part E B C F, but the weight E G H F is taken from it; therefore (by the first Supposition) the parts A G H D and G B C H will not be equally pois'd; and consequently G H is not a Plain of Equi∣ponderation. Wherefore, No two Plains of Equiponderation, &c. Which was to be proved.

3 The Center of Equiponderation is in every Plain of Equi∣ponderation.

For if another Plain of Equiponderation be taken, it will not (by the last Article) be parallel to the former Plain; and therefore both those Plains will cut one another. Now that Section (by the 6th Definition) is the Diameter of Equiponderation. Again, if ano∣ther Diameter of Equiponderation be taken, it will cut that for∣mer Diameter; and in that Section (by the 7th Definition) is the Center of Equiponderation. Wherefore the Center of Equi∣ponderation is in that Diameter which lies in the said Plain of Equiponderation.

4 The Moment of any Ponderant applyed to one point of the Beam, to the Moment of the same, or an equal Ponderant applyed to any other point of the Beam, is as the distance of the former point from the Center of the Scale, to the distance of the later point from the same Center. Or thus, Those Moments are to one another, as the Arches of Circles which are made upon the Cen∣ter of the Scale through those points, in the same time. Or lastly thus; They are, as the parallel bases of two Triangles, which have a common angle at the Center of the Scale.

Let A (in the 2d figure) be the Center of the Scale; and let the equal Poderants D and E press the Beam A B in the points B and C; also let the straight lines B D and C E be Diameters of Equi∣ponderation; and the points D and E in the Ponderants D and E be their Centers of Equiponderation. Let A G F be drawn how∣soever,

cutting D B produced in F, and E C in G; and lastly, upon the common Center A, let the two arches B H and C I be descri∣bed, cutting A G F in H and I. I say the Moment of the Ponderant D to the Moment of the Ponderant E, is as A B to A C, or as B H to C I, or as B F to C G. For the effect of the Ponderant D in the point B, is circular motion in the arch B H; and the effect of the Ponderant E in the point C, circular motion in the arch C I; and by reason of the equality of the Ponderants D and E, these moti∣ons are to one another as the Quicknesses or Velocities with which the points B and C describe the arches B H and C I, that is, as the arches themselves B H and C I, or as the straight parallels B F and C G, or as the parts of the Beam A B and A C; for A B. A C :: B F. C G :: B H. C I. are proportionals; and therefore the effects, that is, (by the 4th Definition) the Moments of the equal Ponde∣rants applyed to several points of the Beam, are to one another, as A B and A C; or as the distances of those points from the center of the Scale; or as the parallel bases of the Triangles which have a common angle at A; or as the concentrick arches B H and C I; which was to be demonstrated.

5 Unequal Ponderants, when they are applyed to several points of the Beam, and hang at liberty (that is, so as the line by which they hang be the Diameter of Equiponderation, whatsoever be the figure of the Ponderant), have their Moments to one another in proportion compounded of the proportions of their distances from the center of the Scale, and of their Waights.

Let A (in the 3d figure) be the center of the Scale, and A B the Beam; to which let the two Ponderants C & D be applied at the points B and E. I say the proportion of the Moment of the Pon∣derant C, to the Moment of the Ponderant D, is compounded of the proportions of A B to A E and of the Waight C to the Waight D; or (if C and D be of the same species) of the magni∣tude C to the magnitude D.

Let either of them, as C, be supposed to be bigger then the o∣ther D. If therefore by the addition of F, F and D together be as one Body equal to C, the Moment of C to the Moment of F + D will be (by the last article) as B G is to E H. Now as F + D is to D, so let E H be to another E I; and the moment of F + D, that is

of C, to the moment of D, will be as B G to E I. But the proporti∣on of B G to E I is compounded of the proportions (of B G to E H that is) of A B to A E, and (of E H to E I, that is) of the waight C to the waight D. Wherefore unequal Ponderants, when they are applied, &c. which was to be proved.

6 The same figure remaining, if I K be drawn parallel to the Beame A B, and cutting A G in K; and K L be drawn parallel to to B G, cutting A B in L, the distances A B and A L from the cen∣ter, will be proportional to the moments of C and D. For the mo∣ment of C is B G, and the moment of D is E I, to which K L is equal. But as the distance A B from the center, is to the distance A L from the center, so is B G the moment of the Ponderant C, to L K, or E I the moment of the Ponderant D.

7 If two Ponderants have their waights and distances from the center in reciprocal proportion, and the center of the Scale be be∣tween the points to which the Ponderants are applied, they will be equally poised. And contrarily, if they be equally poised, their waights and distances from the center of the Scale will be in reci∣procall proportion.

Let the center of the Scale (in the same 3d figure) be A, the Beam A B; and let any Ponderant C, having B G for its moment, be applied to the point B; also let any other Ponderant D, whose moment is E I, be applied to the point E. Through the point I, let I K be drawn parallel to the Beam A B, cutting A G in K; also let K L be drawn parallel to B G. K L will then be the Mo∣ment of the Ponderant D; and (by the last Article) it will be as B G the Moment of the Ponderant C in the point B, to L K the Mo∣ment of the Ponderant D in the point E▪ so A B to A L. On the o∣ther side of the center of the Scale, let A N be taken equal to A L; and to the point N let there be applyed the Ponderant O, having to the Ponderant C the proportion of A B to A N. I say the Ponderants in B and N will be equally poised. For the pro∣portion of the Moment of the Ponderant O in the point N, to the Moment of the Ponderant C in the point B, is (by the 5th Article) cōpounded of the proportions of the waight O to the waight B, & of the distance (from the center of the Scale) A N or A L to the di∣stāce (frō the center of the Scale) A B. But seeing we have supposed, that the distance A B to the distance A N, is in reciprocal propor∣tion

of the Waight O to the waight C, the proportion of the Mo∣ment of the Ponderant O in the point N, to the Moment of the Ponderant C in the point B, will be compounded of the proporti∣ons of A B to A N, and of A N to A B. Wherefore, setting in order A B, A N, A B, the Moment of O to the Moment of C will be as the first to the last, that is, as A B to A B. Their Moments there∣fore are equal; and consequently the Plain which passes through A, will (by the fifth Definition) be a Plain of Equiponderation. Wherefore they will be equally poised; as was to be proved.

Now the converse of this is manifest. For if there be Equipon∣deration, and the proportion of the Waights and Distances be not reciprocal, then both the Waights will alwayes have the same Moments, although one of them have more waight added to it, or its distance changed.

Corollary. When Ponderants are of the same Species, and their Moments be equal; their Magnitudes and Distances from the cen∣ter of the Scale will be reciprocally proportional. For in Homoge∣neous Bodies, it is as Waight to Waight, so Magnitude to Mag∣altude.

8 If to the whole length of the Beam there be applyed a Pa∣rallelogram, or a Parallelopipedum, or a Prisma, or a Cylinder, or the Superficies of a Cylinder, ot of a Sphere, or of any portion of a Sphere or Prisma; the parts of any of them cut off with plains pa∣rallel to the base, will have their Moments in the same proportion with the parts of a Triangle which has its Vertex in the center of the Scale, and for one of its sides the Beam it self, which parts are cut off by Plains parallel to the base.

First, let the rectangled Parallelogram A B C D (in the 4th fi∣gure) be applyed to the whole length of the Beam A B; and pro∣ducing C B howsoever to E, let the Triangle A B E be described. Let now any part of the Parallelogram, as A F, be cut off by the plain F G, parallel to the base C B; and let F G be produced to A E in the point H. I say the Moment of the whole A B C D to the Moment of its part A F, is as the Triangle A B E to the Tri∣angle A G H, that is, in proportion duplicate to that of the distan∣ces from the center of the Scale.

For, the Parallelogram A B C D being divided into equal parts

infinite in number, by straight lines drawn parallel to the base; and supposing the Moment of the straight line C B to be B E; the Moment of the straight line F G, will (by the 7th Arti∣cle) be G H; and the Moments of all the straight lines of that Parallelogram, will be so many straight lines in the Triangle A B E drawn parallel to the base B E; all which parallels together taken are the Moment of the whole Parallelogram A B C D; and the same parallels do also constitute the superficies of the Triangle A B E. Wherefore the Moment of the Parallelogram A B C D, is the Triangle A B E; and for the same reason, the Moment of the Parallelogram A F, is the Triangle A G H; and therefore the Mo∣ment of the whole Parallelogram, to the Moment of a Parallelo∣gram which is part of the same, is as the Triangle A B E, to the Triangle A G H, or in proportion duplicate to that of the Beams to which they are applyed. And what is here demonstrated in the case of a Parallelogram, may be understood to serve for that of a Cylinder, and of a Prisma, and their Superficies; as also for the Superficies of a Sphere, of an Hemisphere, or any portion of a Sphere, (for the parts of the Superficies of a Sphere, have the same proportion with that of the parts of the Axis cut off by the same parallels by which the parts of the Superficies are cut off, as Archimedes has demonstrated); and therefore when the parts of any of these figures are equal and at equal distances from the Center of the Scale, their Moments also are equal, in the same manner as they are in Parallelograms.

Secondly, let the Parallelogram A K I B not be rectangled; the straight line I B wil nevertheless press the point B perpendicularly in the straight line B E; & the straight line L G wil press the point G perpendicularly in the straight line G H; and all the rest of the straight lines which are parallel to I B will do the like. Whatsoe∣ver therefore the Moment be which is assigned to the straight line I B, as here (for example) it is supposed to be B E, if A E be drawn, the Moment of the whole Parallelogram A I will be the Triangle A B E; and the Moment of the part A L will be the Triangle A G H. Wherefore the Moment of any Ponderant, which has its sides equally applyed to the Beam, (whether they be applyed per∣pendicularly or obliquely) will be always to the Moment of a part of the same, in such proportion, as the whole Triangle has to a part

of the same cut off by a plain which is parallel to the base.

9 The Center of Equiponderation of any figure which is defi∣cient according to commensurable proportions of the altitude and base diminished, and whose complete figure is either a Parallelo∣gram, or a Cylinder, or a Parallelopipedum, divides the Axis, so, that the part next the Vertex, to the other part, is as the complete fi∣gure to the deficient figure.

For let C I A P E (in the 5th figure) be a deficient figure, whose Axis is A B, and whose complete figure is C D F E; and let the Axis A B be so divided in Z, that A Z be to Z B as C D F E is to C I A P E. I say the center of Equiponderation of the figure C I A P E will be in the point Z.

First, that the Center of Equiponderation of the figure C I A∣P E is somewhere in the Axis A B, is manifest of it self; and there∣fore A B is a Diameter of Equiponderation. Let A E be drawn, and let B E be put for the Moment of the straight line C E; the Triangle A B E will therefore (by the 3d Article) be the Moment of the complete figure C D F E. Let the Axis A B be equally di∣vided in L, and let G L H be drawn parallel and equal to the straight line C E, cutting the crooked line C I A P E in I and P, and the straight lines A C and A E in K and M. Moreover, let Z O be drawn parallel to the same C E; and let it be, as L G to L I, so L M to another L N; and let the same be done in all the rest of the straight lines possible, parallel to the base; and through all the points N, let the line A N E be drawn; the three-sided figure A N E B will therefore be the Moment of the figure C I A P E. Now the Triangle A B E is (by the 9th Article of the 17th Chap∣ter) to the three-sided figure A N E B, as A B C D + A I C B is to A I C B twice taken, that is, as C D F E + C I A P E is to C I A P E twice taken. But as C I A P E is to C D F E, that is, as the waight of the deficient figure, is to the waight of the complete figure, so is C I A P E twice taken, to C D F E twice taken. Wherefore, setting in order C D F E + C I A P E. 2 C I A P E. 2 C D F E; the proportion of C D F E + C I A P E to C D F E twice taken, will be compounded of the proportion of C D F E + C I A P E to C I A P E twice taken, that is, of the pro∣portion of the Triangle A B E to the threesided figure A N E B, that is, of the Moment of the complete figure to the Moment of

the deficient figure, and of the proportion of C I A P E twice ta∣ken, to C D F E twice taken, that is, to the proportion reciprocally taken of the waight of the deficient figure to the waight of the complete figure.

Again, seeing by supposition A Z. Z B :: C D F E. C I A P E are proportionals; A B. A Z :: C D F E + C I A P E. C D F E will also (by cōpounding) be proportionals. And seeing A L is the half of A B, A L. A Z :: C D F E + C I A P E. 2 C D F E will also be pro∣portionals. But the proportion of C D F E + C I A P E to 2 C D F E is compounded (as was but now shewn) of the proportions of Mo∣ment to Moment &c. and therefore the proportion of A L to A Z is compounded of the proportion of the Moment of the complete figure C D F E to the Moment of the deficient figure C I A P E, and of the proportion of the waight of the deficient figure C I A∣P E, to the waight of the complete figure C D F E; But the pro∣portion of A L to A Z is compounded of the proportions of A L to B Z and of B Z to A Z. Now the proportion of B Z to A Z is the proportion of the Waights reciprocally taken, that is to say, of the waight C I A P F to the waight C D F E. Therefore the remayning proportion of A L to B Z, that is, of L B to B Z is the proportion of the Moment of the waight C D F E to the Moment of the waight C I A P E. But the proportion of A L to B Z is com∣pounded of the proportions of A L to A Z and of A Z to Z B; of which proportions that of A Z to Z B is the proportion of the waight C D F E to the waight C I A P E. Wherefore (by the 5th Article of this Chapter) the remayning proportion of A L to A Z is the proportion of the distances of the points Z and L from the center of the Scale, which is A. And therefore (by the 6th Article) the waight C I A P E shall hang from O in the straight line O Z. So that O Z is one Diameter of Equiponderation of the waight C I A P E. But the straight line A B is the other Dia∣meter of Equiponderation of the same waight C I A P E. Where∣fore (by the 7th Definition) the point Z is the center of the same Equiponderation; which point (by construction) divides the axis so, that the part A Z which is the part next the vertex, is to the other part Z B, as the complete figure C D F E is to the deficient figure C I A P E; which is that which was to be demonstra∣ted.

• Corollary. The Center of Equiponderation of any of those plain three-sided figures, which are compared with their complete fi∣gures in the Table of the third Article of the 17th Chapter, is to be found in the same Table, by taking the Denominator of the fra∣ction for the part of the axis cut off next the vertex, and the Nu∣merator for the other part next the base. For example, if it be re∣quired to find the Center of Equiponderation of the second three∣sided figure of foure Meanes, there is in the concourse of the se∣cond columne with the row of three-sided figures of four Meanes this fraction ••/7, which signifies that that figure is to its parallelogrā or compleat figure as 5/7 to Unity, that is, as 5/7 to 7/7, or as 5 to 7; and therefore the Center of Equiponderation of that figure, divides the axis, so, that the part next the vertex is to the other part as 7 to 5.
• 2 Corallary. The Center of Equiponderation of any of the So∣lids of those figures which are contained in the Table of the 8th Article of the same 17th Chapter, is exhibited in the same Table. For example, if the Center of Equiponderation of a Cone be sought for; the Cone will be found to be ⅓ of its Cylinder; and therefore the Center of its Equiponderation will so divide the axis, that the part next the vertex, to the other part, will be as 3 to 1. Also the Solid of a three-sided figure of one Meane, that is, a parabolical Solid, seeing it is 2/4, that is ½ of its Cylinder, will have its Center of Equiponderation in that point, which di∣vides the axis, so, that the part towards the vertex be double to the part towards the base.

10 The Diameter of Equiponderation of the Complement of the half of any of those figu••es which are contained in the Table of the 3d article of the 17th Chapter, divides that line which is drawne through the Vertex parallel and equall to the base, so, that the part next the Vertex, will be to the other part, as the Complete figure to the Complement.

For let A I C B (in the same 5 fig.) be the halfe of a Parabola, or of any other of those three-sided figures which are in the Table of the 3d article of the 17th Chap whose Axis is A B, and base B C; having A D drawn from the Vertex, equall and parallel to the base B C; and whose complete figure is the parallelogramme A B C D. Let I Q be drawne, at any distance from the side C D,

but parallel to it; and let A D be the altitude of the Complement A I C D, and Q I a line ordinately applyed in it. Wherefore the altitude A L in the deficient figure A I C B, is equal to Q I the line ordinately applyed in its Complement; and contrarily, L I the line ordinately applyed in the figure A I C B, is equall to the altitude A Q in its Complement; and so in all the rest of the ordinate lines and altitudes, the mutation is such, that that line which is ordinately applyed in the figure, is the altitude of its Complement. And therefore the proportion of the altitudes decreasing, to that of the ordinate lines decrea∣sing, being multiplicate according to any number in the deficient figure, is submultiplicate according to the same number in its Complement. For example, if A I C B be a Parabola, seeing the proportion of A B to A L is duplicate to that of B C to L I, the proportion of A D to A Q in the Complement A I C D (which is the same with that of B C to L I) will be subduplicate to that of C D to Q I (which is the same with that of A B to A L); and con∣sequently, in a Parabola, the Complement will be to the Paral∣lelogramme as 1 to 3; in a three-sided figure of two Meanes, as 1 to 4; in a three-sided figure of three Meanes, as 1 to 5, &c. But all the ordinate lines together in A I C D are its moment; and all the ordinate lines in A I C B are its moment. Wherefore the moments of the Complements of the halves of Deficient figures in the Table of the 3d article of the 17th Chap. being compared, are as the Deficient figures themselves; and therefore the Dia∣meter of Equiponderation will divide the straight line A D in such proportion, that the part next the Vertex be to the other part, as the complete figure A B C D is to the Complement A I C D.

Coroll. The diameter of Equiponderation of these halves, may be found by the Table of the ••d article of the 17th Chapter in this manner. Let there be propounded any deficient figure, namely the second three-sided figure of two Meanes. This figure is to the complete figure as ⅗ to 1, that is as 3 to 5. Wherefore the Com∣plement to the same complete figure is as 2 to 5; and therefore the diameter of Equiponderation of this Complement will cut the straight line drawne from the Vertex parallel to the base, so, that the part next the Vertex will be to the other part as 5 to 2.

And in like manner, any other of the said three-sided figures being propounded, if the numerator of its fraction (found out in the Table) be taken from the denominator, the straight line drawn from the Vertex is to be divided, so, that the part next the Vertex be to the other part, as the denominator is to the remain∣der which that substraction leaves.

11 The center of Equiponderation of the halfe of any of those crooked-lined figures which are in the first row of the Table of the 3d article of the 17th chapter, is in that straight line, which being parallel to the Axis, divides the base according to the num∣bers of the fraction next below it in the second row, so, that the Numerator be answerable to that part which is towards the Axis.

For example, let the first figure of three Means be taken, whose half is A B C D (in the 6th figure), and let the rectangle A B E D be completed. The Complement therefore will be B C D E. And seeing A B E D is to the figure A B C D (by the Table) as 5 to 4, the same A B E D will be to the Complement B C D E as 5 to 1. Wherefore if F G be drawn parallel to the base D A, cutting the axis, so, that A G be to G B as 4 to 5, the cen∣ter of Equiponderation of the figure A B C D, will (by the prece∣dent article) be somewhere in the same F G. Again, seeing (by the same article) the complete figure A B E D, is to the Comple∣ment B C D E as 5 to 1, therefore if B E and A D be divided in H and I as 5 to 1, the center of Equiponderation of the Comple∣ment B C D E will be somewhere in the straight line which con∣nects H and I. Let now the straight line L K be drawn through M the center of the complete figure, parallel to the base; and the straight line N O, through the same center M, perpen∣dicular to it; and let the straight lines L K and F G cut the straight line H I in P and Q. Let P R be taken quadruple to P Q; and let R M be drawn and produced to F G in S. R M therefore will be to M S as 4 to 1, that is, as the figure A B C D to its Comple∣ment B C D E. Wherefore seeing M is the center of the Com∣plete figure A B E D, and the distances of R and S from the cen∣ter M be in proportion reciprocall to that of the waight of the Complement B C D E to the waight of the figure A B C D, R and S will either be the centers of Equiponderation of their own

figures, or those centers will be in some other points of the diame∣ters of Equiponderation H I and F G. But this last is impossible. For no other straight line can be drawn through the point M ter∣minating in the straight lines H I and F G, and retaining the pro∣portion of M R to M S, that is, of the figure A B C D to its comple∣ment B C D E. The center therefore of Equiponderation of the figure A B C D is in the point S. Now seeing P M hath the same proportion to Q S which R P hath to R Q, Q S will be 5 of those parts of which P M is 4, that is, of which I N is 4. But I N or P M is 2 of those parts of which E B or F G is 6; and therefore if it be, as 4 to 5, so 2 to a fourth, that fourth will be 2½. Wherefore Q S is 2½ of those parts of which F G is 6. But F Q is 1; and there∣fore F S is 3½. Wherefore the remayning part G S is 2½. So that F G is so divided in S, that the part towards the Axis, is in proportion to the other part as 2½ to 3½, that is, as 5 to 7; which answereth to the fraction 5/7 in the second row, next under the fraction ⅘ in the first row. Wherefore drawing S T parallel to the Axis, the base wil be divided in like manner.

By this Method it is manifest, that the base of a Semiparabola will be divided into 3 and 5; and the base of the first three-sided figure of two Means, into 4 and 6; and of the first three-sided figure of four Means, into 6 and 8. The fractions therefore of the second row denote the proportions into which the bases of the figures of the first row are divided by the diameters of Equipon∣deration. But the first row begins one place higher then the second row.

12 The center of Equiponderation of the half of any of the fi∣gures in the second row of the same Table of the 3d article of the 17th Chapter, is in a straight line parallel to the Axis, and divi∣ding the base according to the nūbers of the fraction in the fourth row, two places lower, so, as that the Numerator be answerable to that part which is next the Axis.

Let the half of the second three-sided figure of two Means be taken; and let it be A B C D (in the 7th Figure); whose comple∣ment is B C D E, and the rectangle completed A B E D. Let this rectangle be divided by the two straight lines L K & N O, cutting one another in the center M at right angles; and because A B E D

is to A B C D as 5 to 3, let A B be divided in G, so, that A G to B G be as 3 to 5; and let F G be drawn parallel to the base. Also be∣cause A B E D is (by the 9th article) to B C D E as 5 to 2, let B E be divided in the point I, so, that B I be to I E as 5 to 2; and let I H be drawn parallel to the Axis, cutting L K and F G in P and Q. Let now P R be so taken, that it be to P Q as 3 to 2, and let R M be drawn and produced to F G in S. Seeing therefore R P is to R Q, that is, R M to M S, as A B C D is to its complement B C D E, and the centers of Equiponderation of A B C D and B C D E are in the straight lines F G and H I, and the center of Equiponderation of them both together in the point M; R will be the center of the Complement B C D E, and S the center of the Figure A B C D. And seeing P M, that is I N, is to Q S, as R P is to R Q; and I N, or P M is 3 of those parts, of which B E, that is, F G is 14; there∣fore Q S is 5 of the same parts; and E I, that is F Q, 4; and F S, 9; and G S, 5. Wherefore the straight line S T being drawn parallel to the Axis, will divide the base A D into 5 and 9. But the fraction 5/9 is found in the fourth row of the Table, two places below the fracti∣on 9/5 in the second row.

By the same method, if in the same second row, there be taken the second three-sided Figure of three Meanes, the center of Equi∣ponderation of the half of it, will be found to be in a straight line parallel to the Axis, dividing the base according to the numbers of the fraction 6/10, two places below in the fourth row. And the same way serves for all the rest of the Figures in the second row. In like manner, the center of Equiponderation of the third three-sided Figure of three Means, will be found to be in a straight line paral∣lel to the Axis, dividing the base, so, that the part next the Axis, be to the other part, as 7 to 13, &c.

Coroll. The Centers of Equiponderation of the halves of the said Figures are known, seeing they are in the intersection of the straight lines S T and F G, which are both known.

13 The center of Equiponderation of the half of any of the Fi∣gures, which (in the Table of the 3d Article of the 17th Chap.) are compared with their Parallelograms, being known; the center of Equiponderation of the excess of the same Figure above its trian∣gle, is also known.

For example, let the Semiparabola A B C D (in the 8th Fig.) be taken; whose Axis is A B; whose complete Figure is A B E D; and whose excess above its triangle is B C D B. Its center of E∣quiponderation may be found out in this manner. Let F G be drawn parallel to the base, so, that A F be a third part of the Axis; and let H I be drawn parallel to the Axis, so, that A H be a third part of the base. This being done, the center of Equiponderation of the trian∣gle A B D, will be I. Again, let K L be drawn parallel to the base, so, that A K be to A B as 2 to 5; and M N parallel to the Axis, so, that A M be to A D as 3 to 8; and let M N terminate in the straight line K L. The center therefore of Equiponderation of the Para∣bola A B C D is N; and therefore we have the centers of Equi∣ponderation of the Semiparabola A B C D, and of its part the triangle A B D. That we may now finde the Center of Equipon∣deration of the remayning part B C D B, let I N be drawn and produced to O, so, that N O be triple to I N; and O will be the center sought for. For seeing the waight of A B D, to the waight of B C D B is in proportion reciprocall to that of the straight line N O to the straight line I N; and N is the center of the whole, and I the center of the triangle A B D; O will be the center of the remaining part, namely, of the figure B D C B; which was to be found.

Coroll. The Center of Equiponderation of the figure B D C B, is in the concourse of two straight lines, whereof one is paral∣lel to the base, and divides the Axis, so, that the part next the base be ⅖ or 6/15 of the whole Axis; the other is parallel to the Ax∣is, and so divides the base, that the part towards the Axis be ½ or 12/24 of the whole base. For drawing O P parallel to the base, it will be as I N to N O, so F K to K P, that is, so 1 to 3, or 5 to 15. But A F is 5/15 or ⅓ of the whole A B; and A K is 6/15 or ⅖; and F K ••/15; and K P 3/15; and therefore A P is 9/15 of the Axis A B. Also A H is ⅓ or 8/24; and A M ⅜ or 9/24 of the whole base; and therefore O Q being drawn parallel to the Axis, M Q (which is triple to H M) will be 3/24. Wherefore A Q is 12/24 or ½ of the base A D.

The excesses of the rest of the three-sided figures in the first row of the Table of the 3d article of the 17th Chapter, have their centers of Equiponderation in two straight lines which divide the

Axis and base according to those fractions, which adde 4 to the numerators of the fractions of a Parabola 9/15 and 12/24; and 6 to the denominators, in this manner,

• In a Parabola, The Axis 9/15, The Base 12/24
• In the first three-sided figure, The Axis 13/21, The Base 16/30
• In the second three-sided figure, The Axis 17/27, The Base 20/36 &c.

And by the same method, any man (if it be worth the paines) may find out the centers of Equiponderation of the excesses a∣bove their triangles of the rest of the figures in the second & third row, &c.

14 The center of Equiponderation of the Sector of a Sphere (that is, of a figure compounded of a right Cone whose Vertex is the center of the Sphere, and the portion of the Sphere whose base is the same with that of the Cone), divides the straight line which is made of the Axis of the Cone and halfe the Axis of the por∣tion together taken, so, that the part next the Vertex be triple to the other part, or to the whole straight line, as 3 to 4.

For let A B C (in the 9th fig.) be the Sector of a Sphere, whose Vertex is the ce••ter of the Sphere A; whose Axis is A D; and the circle upon B C is the common base of the portion of the Sphere and of the Cone whose Vertex is A; the Axis of which portion is E D, and the halfe thereof F D; and the Axis of the Cone, A E. Lastly let A G be ¾ of the straight line A F. I say G is the center of Equiponderation of the Sector A B C.

Let the straight line F H be drawne of any length, making right angles with A F at F; and drawing the straight line A H, let the triangle A F H be made. Then upon the same center A let any arch I K be drawne, cutting A D in L; and its chord, cutting A D in M; and dividing M L equally in N, let N O be drawne parallel to the straight line F H, and meeting with the straight line A H in O.

Seeing now B D C is the Spherical Superficies of the portion cut off with a plain passing through B C, and cutting the Axis at right angles; and seeing F H divides E D the Axis of the portion into two equal parts in F; the center of Equiponderation of the Super∣ficies

B D C will be in F (by the 8th article); and for the same reason the center of Equiponderation of the Superficies I L K (K being in the straight line A C) will be in N. And in like manner, if there were drawne between the center of the Sphere A and the outermost Spherical Superficies of the Sector, arches infinite in number, the centers of Equiponderation of the Sphericall Su∣perficies in which those arches are,, would be found to be in that part of the Axis, which is intercepted between the Superfi∣cies it selfe and a plaine passing along by the chord of the arch, and cutting the Axis in the middle at right angles.

Let it now be supposed that the moment of the outermost sphe∣ricall Superficies B D C is F H. Seeing therefore the Superficies B D C is to the Superficies I L K in proportion duplicate to that of the arch B D C to the arch I L K, that is, of B E to I M, that is, of F H to N O; let it be as F H to N O, so N O to another N P; and again, as N O to N P, so N P to another N Q; and let this be done in all the straight lines parallel to the base F H that can possibly be drawn between the base and the vertex of the triangle A F H. If then through all the points Q there be drawn the crooked line A Q H, the figure A F H Q A will be the complement of the first three-si••ed figure of two Meanes; and the same will also be the moment of all the Sphericall Super∣ficies of which the Solid Sector A B C D is compounded; and by consequent, the moment of the Sector it selfe. Let now F H be understood to be the semidiameter of the base of a right Cone, whose side is A H, and Axis A F. Wherfore seeing the bases of the Cones which passe through F and N and the rest of the points of the Axis, are in proportion duplicate to that of the straight lines F H and N O, &c. the moment of all the bases together, that is, of the whole Cone, will be the figure it self A F H Q A; and there∣fore the center of Equiponderation of the Cone A F H is the same with that of the solid Sector. Wherefore seeing A G is ¾ of the Axis A F, the center of Equiponderation of the Cone A F H is in G; and therefore the center of the solid Sector is in G also, and divides the part A F of the Axis, so, that A G is triple to G F; that is, A G is to A F as 3 to 4; which was to be demonstrated.

Note, that when the Sector is a Hemisphere, the Axis of the

Cone vanisheth into that point which is the center of the Sphere; and therefore it addeth nothing to half the Axis of the portion. Wherefore, if in the Axis of the Hemisphere, there be taken from the center, ¾ of halfe the Axis, that is, 3/•• of the Semidiame∣ter of the Sphere, there will be the center of Equiponderation of the Hemisphere.

CHAP. XXIV. Of Refraction and Reflection.

• 1 Definitions.
• 2 In perpendicular Motion there is no Refraction.
• 3 Things thrown out of a thinner into a thicker Medium, are so refracted, that the Angle Refracted is greater then the Angle of Inclination.
• 4 En∣deavour which from one point tendeth every way, will be so Refracted, at that the sine of the Angle Refracted, will be to the sine of the Angle of Inclination, as the Density of the first Medium is to the Density of the second Medium, reciprocally taken.
• 5 The sine of the Refracted Angle in one Inclination is to the sine of the Refracted Angle in another Inclination, as the sine of the Angle of that Inclination is to the sine of the Angle of this Inclination.
• 6 If two lines of Incidence, having equal Inclination, be the one in a thinner, the other in a thicker Medium, the sine of the angle of Inclina∣tion will be a Mean proportional between the two sines of the Refracted angles.
• 7 If the angle of Inclination be semirect, and the line of Inclination be in the thicker Medium, and the proportion of their Densities be the same with that of the Diagonal to the side of a Square, and the separating Superficies be plain, the Refracted line will be in the separating Superficies.
• 8 If a Body be carried in a straight line upon another Body, and do not penetrate the same, but be reflected from it, the angle of Reflexion will be equal to the Angle of Incidence.
• 9 The same happens in the generation of Motion in the line of Incidence.

1

Definitions.

• 1 REFRACTION, is the breaking of that straight Line, in which a Body is moved, or its Action would proceed in one and the same Medium, into two straight lines, by reason of the different natures of the two Mediums.
• 2 The former of these is called the Line of Incidence; the later the Refracted Line.
• 3 The Point of Refraction, is the common point of the Line of In∣cidence and of the Refracted Line.
• ...

• 4 The Refracting Superficies, which also is the Separating Superficies of the two Mediums, is that in which is the point of Refra∣ction.
• 5 The Angle Refracted, is that which the Refracted Line makes in the point of Refraction, with that Line which from the same point is drawn perpendicular to the separating Superfi∣cies in a different Medium.
• 6 The Angle of Refraction, is that which the Refracted line makes with the Line of Incidence produced.
• 7 The Angle of Inclination, is that which the Line of Incidence makes with that Line which from the point of Refraction is drawn perpendicular to the separating Superficies.
• ...

  8 The Angle of Incidence, is the Complement to a right Angle of the Angle of Inclination.

  And so, (in the first Figure) the Refraction is made in A B F. The Refracted Line is B F. The Line of Incidence is A B. The Point of Incidence, and of Refraction is B. The Refracting or Separating Su∣perficies is D B E. The Line of Incidence produced directly is A B C The Perpendicular to the separating Superficies is B H. The Angle of Refraction is C B F. The Angle Refracted is H B F. The Angle of Inclination is A B G or H B C. The Angle of Inci∣dence is A B D.

• 9 Moreover the Thinner Medium, is understood to be that in which there is less resistance to Motion or to the generation of Motion; & the Thicker, that wherin there is greater resistance.
• 10 And that Medium in which there is equal resistance every where, is a Homogeneous Medium. All other Mediums are Hete∣rogeneous.

2 If a Body pass, or there be generation of Motion, from one Medium to another of different Density, in a line perpendicular to the Separating Superficies; there will be no Refraction.

For seeing on every side of the perdendicular all things in the Mediums are supposed to be like and equal; if the Motion it self be supposed to be perpendicular, the Inclinations also will be e∣qual, or rather none at all; and therefore there can be no cause, from which Refraction may be inferred to be on one side of the

perpendicular, which wil not cōclude the same Refraction to be on the other side. Which being so, Refraction on one side will destroy Refraction on the other side; and consequently, either the Refra∣cted line will be every where, (which is absurd), or there will be no Refracted line at all; which was to be demonstrated.

Corol. It is manifest from hence, that the cause of Refraction con∣sisteth onely in the obliquity of the line of Incidence, whether the Incident Body penetrate both the Mediums, or without penetra∣ting, propagate motion by Pressure onely.

3 If a Body, without any change of situation of its internal parts, as a stone, be moved obliquely out of the thinner Medium, and proceed penetrating the thicker Medium; and the thicker Me∣dium be such, as that its internal parts being moved, restore them∣selves to their former situation; the angle Refracted will be greater then the angle of Inclination.

For let D B E (in the same first figure) be the separating Super∣ficies of two Mediums; and let a Body, as a stone thrown, be un∣derstood to be moved as is supposed in the straight line A B C; and let A B be in the thinner Medium, as in the Aire; and B C in the thicker, as in the Water. I say the stone, w^ch being thrown, is moved in the line A B, will not proceed in the line B C, but in some other line, namely that, with which the perpendicular B H makes the Refracted angle H B F greater then the angle of Inclination H B C.

For seeing the stone coming from A, and falling upon B, makes that which is at B proceed towards H, and that the like is done in all the straight lines which are parallel to B H; and seeing the parts moved restore themselves by contrary motion in the same line; there will be contrary motion generated in H B, and in all the straight lines which are parallel to it. Wherefore the motion of the stone will be made by the concourse of the motions in A G, that is, in D B, and in G B, that is, in B H, and lastly, in H B, that is, by the concourse of three motions. But by the concourse of the motions in A G and B H, the stone will be carried to C; and therefore by ad∣ding the motion in H B, it will be carried higher in some other line, as in B F, and make the angle H B F greater then the angle H B C.

And from hence may be derived the cause, why Bodies which

are thrown in a very oblique line, if either they be any thing flat, or be thrown with great force, will when they fall upon the water, be cast up again from the water into the aire.

For let A B (in the 2d figure) be the superficies of the water; in∣to which from the point C, let a stone be thrown in the straight line C A, making with the line B A produced a very little angle C A D; and producing B A indefinitely to D, let C D be drawn perpendi∣cular to it, and A E parallel to C D. The stone therefore will be moved in C A by the concourse of two motions in C D and D A, whose velocities are as the lines themselves C D and D A. And from the motion in C D and all its parallels downwards, as soon as the stone falls upon A, there will be Reaction upwards, be∣cause the water restores it self to its former situation. If now the stone be thrown with sufficient obliquity, that is, if the straight line C D be short enough, that is, if the endeavour of the stone downwards be less then the Reaction of the water upwards, that is, less then the endeavour it hath from its own gravity, (for that may be), the stone will (by reason of the excess of the endeavour which the water hath to restore it self, above that which the stone hath downwards) be raised again above the Superficies A B, and be carried higher, being reflected in a line which goes higher, as the line A G.

4 If from a point, whatsoever the Medium be, Endeavour be propagated every way into all the parts of that Medium; and to the same Endeavour there be obliquely opposed another Medium of a different nature, that is, either thinner or thicker; that Endea∣vour will be so refracted, that the sine of the angle Refracted, to the sine of the angle of Inclination, will be as the density of the first Medium to the density of the second Medium, reciprocally ta∣ken.

First, let a Body be in the thinner Medium in A (Figure 3d.); and let it be understood to have endeavour every way, and conse∣quently that its endeavour proceed in the lines A B and A b; to which let B b the superficies of the thicker Medium be obliquely opposed in B and b, so that A B and A b be equal; and let the straight line B b be produced both wayes. From the points B and b let the perpendiculars B C and b c be drawn; and upon the centers

B and b, and at the equal distances B A and b A, let the Circles A C and A c be described, cutting B C and b c in C and c, and the same C B and c b produced in D and d, as also A B and A b pro∣duced in E and e. Then from the point A to the straight lines B C and b c let the perpendiculars A F and A f be drawn. A F therefore will be the sine of the angle of Inclination of the straight line A B, and A f the sine of the angle of Inclination of the straight line A h, which two Inclinations are by construction made equal. I say, as the density of the Medium in which are B C and b c, is to the density of the Medium in which are B D and b d, so is the sine of the angle Refracted, to the sine of the angle of Inclination.

Let the straight line F G be drawn parallel to the straight line A B, meeting with the straight line b B produced in G.

Seeing therefore A F and B G are also parallels, they will be e∣qual; and consequently, the endeavour in A F is propagated in the same time, in which the endeavour in B G would be propagated if the Medium were of the same density. But because B G is in a thicker Medium, that is, in a Medium which resists the endeavour more then the Medium in which A F is, the endeavour will be propagated less in B G then in A F, according to the propor∣tion which the density of the Medium in which A F is, hath to the density of the Medium in which B G is. Let therefore the density of the Medium in which B G is, be to the density of the Medium in which A F is, as B G is to B H; and let the measure of the time be the Radius of the Circle. Let H I be drawn parallel to B D, meeting with the circumference in I; and from the point I let I K be drawn perpendicular to B D; which being done, B H and I K will be equal; and I K will be to A F, as the density of the Medium in which is A F, is to the density of the Medium in which is I K. Seeing therefore in the time A B (which is the Radius of the Circle) the endeavour is propagated in A F in the thinner Me∣dium, it will be propagated in the same time, that is, in the time B I in the thicker Medium from K to I. Therefore B I is the Refra∣cted line of the line of Incidence A B; and I K is the sine of the angle Refracted; and A F, the sine of the angle of Inclination. Wherefore seeing I K is to A F, as the density of the Medium in which is A F to the density of the Medium in which is I K; it will be as the density of the Medium in which is A F, (or

B C) to the density of the Medium in which is I K (or B D), so the sine of the angle Refracted to the sine of the angle of Inclination. And by the same reason it may be shewn, that as the density of the thinner Medium is to the density of the thicker Medium, so will K I the sine of the angle Refracted be to A F the sine of the Angle of Inclination.

Secondly, let the Body which endeavoureth every way, be in the thicker Medium at I. If therefore both the Mediums were of the same density, the endeavour of the Body in I B would tend di∣rectly to L; and the sine of the angle of Inclination L M would be equal to I K or B H. But because the density of the Medium in which is IK, to the density of the Medium in which is L M, is as BH to B G, that is, to A F, the endeavour will be propagated further in the Medium in which L M is, then in the Medium in which I K is, in the proportion of density to density, that is, of M L to A F. Wherefore B A being drawn, the angle Refracted will be C B A, and its sine A F. But L M is the sine of the angle of Inclination; and therefore again, as the density of one Medium is to the densi∣ty of the different Medium, so reciprocally is the sine of the angle Refracted to the sine of the angle of Inclination, which was to be demonstrated.

In this Demonstration, I have made the separating Superficies B b plain by construction. But though it were concave or convex, the Theoreme would nevertheless be true. For the Refraction be∣ing made in the point B of the plain separating Superficies, if a crooked line, as P Q be drawn, touching the separating line in the point B; neither the Refracted line B I, nor the perpendicular B D will be altered; and the Refracted angle K B I, as also its sine K I will be still the same they were.

5 The sine of the angle Refracted in one Inclination, is to the sine of the angle Refracted in another Inclination, as the sine of the angle of that Inclination to the sine of the angle of this Incli∣nation.

For seeing the sine of the Refracted angle is to the sine of the angle of Inclination, (whatsoever that Inclination be) as the density of one Medium, to the density of the other Medium; the proporti∣on of the sine of the Refracted angle, to the sine of the angle of In∣clination, will be compounded of the proportions of density to den∣sity,

and of the sine of the angle of one Inclination to the sine of the angle of the other Inclination. But the proportions of the densities in the same Homogeneous Body, are supposed to be the same. Wherefore Refracted angles in different Inclinations, are as the sines of the angles of those Inclinations; which was to be demon∣strated.

6 If two lines of Incidence having equal inclination, be the one in a thinner, the other in a thicker Medium; the sine of the angle of their Inclination, will be a mean proportional between the two sines of their angles Refracted.

For let the straight line AB (in the same 3d figure) have its Incli∣nation in the thinner Medium, and be refracted in the thicker Me∣dium in B I; and let E B have as much Inclination in the thicker Medium, and be refracted in the thinner Medium in B S; and let R S the sine of the angle Refracted be drawn. I say the straight lines R S, A F and I K are in continual proportion. For it is, as the density of the thicker Medium to the density of the thinner Me∣dium, so R S to A F. But it is also, as the density of the same thicker Medium, to that of the same thinner Medium, so AF to IK. Where∣fore R S. A F : : A F. I K are propoortionals; that is, R S, A F and I K are in continual proportion, and A F is the Mean proportio∣nal; which was to be proved.

7 If the angle of Inclination be semirect, and the line of Incli∣nation be in the thicker Medium, and the proportion of the Densi∣ties be as that of a Diagonal to the side of its Square, and the separating Superficies be plain, the Refracted line will be in that separating Superficies.

For in the Circle A C (in the 4th figure) let the angle of Incli∣nation A B C be an angle of 45 degrees. Let C B be produced to the Circumference in D; & let C E (the sine of the angle •• B C) be drawn▪ to which, let B F be taken equal in the separating line B G. B C E F will therefore be a Parallelogram, & F E & B C, that is, F E and B G equal. Let AG be drawn, namely, the Diagonal of the Square whose side is B G; and it will be, as A G to E F, so B G to B F; & so (by supposition) the density of the Medium in which C is, to the density of the Medium in which D is; and so also the sine of the angle Refracted to the sine of the angle of Inclination.

Drawing therefore F D, & from D the line D H perpendicular to A B produced, DH will be the sine of the angle of Inclination. And seeing the sine of the angle Refracted is to the sine of the angle of Inclination, as the density of the Medium in which is C, is to the density of the Medium in which is D, that is, (by supposition) as A G is to F E, that is, as D H is to B G; and seeing D H is the sine of the angle of Inclination, B G will therefore be the sine of the angle Refracted. Wherefore B G will be the Refracted line, and lye in the plain separating Superficies; which was to be demonstrated.

Coroll. It is therefore manifest, that when the Inclination is greater then 45 degrees, as also when it is less, provided the densi∣ty be greater, it may happen that the Refraction will not enter the thinner Medium at all.

8 If a Body fall in a straight line upon another Body, and do not penetrate it, but be reflected from it, the angle of Reflexion will be equal to the angle of Incidence.

Let there be a Body at A (in the 5th figure), which falling with straight motion in the line A C upon another Body at C, passeth no further, but is reflected; and let the angle of Incidence be any angle, as A C D. Let the straight line C E be drawn, making with D C produced the angle E C F equall to the angle A C D; and let A D be drawn perpendicular to the straight line D F. Also in the same straight line D F let C G be taken equall to C D; and let the perpendicular G E be raised, cutting C E in E. This be∣ing done, the triangles A C D and E C G will be equall and like. Let C H be drawn equal and parallel to the straight line A D; and let H C be produced indefinitely to I. Lastly let E A be drawn, which will passe through H, and be parallel and equall to G D. I say the motion from A to C in the straight line of Inci∣dence AC, will be reflected in the straight line C E.

For the motion from A to C is made by two coefficient or con∣current motions, the one in A H parallel to D G, the other in A D perpendicular to the same D G; of which two motions, that in A H workes nothing upon the Body A after it has been moved as farre as C, because (by supposition) it doth not passe the straight line D G; whereas the endeavour in A D, that is in H C, work∣eth further towards I. But seeing it doth onely presse and not pe∣netrate,

there will be reaction in H, which causeth motion from C towards H; and in the mean time the motion in H E re∣maines the same it was in A H; and therefore the Body will now be moved by the concourse of two motions in C H and H E, which are equall to the two motions it had formerly in A H and H C. Wherefore it will be carried on in C E. The angle therefore of Re∣flection will be E C G, equall (by construction) to the angle A C D; which was to be demonstrated.

Now when the Body is considered but as a point, it is all one, whether the Superficies or line in which the Reflection is made, be straight or crooked; for the point of Incidence and Reflexion C, is as well in the crooked line which toucheth D G in C, as in D G it selfe.

9 But if we suppose that not a Body be moved, but some Endea∣vour onely be propagated from A to C, the Demonstration will neverthelesse be the same. For all Endeavour is motion; and when it hath reached the Solid Body in C, it presseth it, and endeavoureth further in C I. Wherefore the reaction will pro∣ceed in C H; and the endeavour in C H concurring with the en∣deavour in H E, will generate the endeavour in C E, in the same manner as in the repercussion of Bodies moved.

If therefore Endeavour be propagated from any point to the concave Superficies of a Spherical Body, the Reflected line with the circumference of a great circle in the same Sphere, will make an angle equall to the angle of Incidence.

For if Endeavour be propagated from A (in the 6 fig.) to the circumference in B, and the center of the Sphere be C, and the line C B be drawne, as also the Tangent D B E; and lastly if the angle F B D be made equall to the angle A B E, the Reflexion will be made in the line B F, as hath been newly shewn. Where∣fore the angles which the straight lines A B and F B make with the circumference, will also be equall. But it is here to be noted that if C B be produced howsoever to G, the endeavour in the line G B C will proceed onely from the perpendicular reaction in G B; and that therefore there will be no other endeavour in the point B towards the parts which are within the Sphere, besides that which tends towards the center.

And here I put an end to the third part of this Discourse; in which I have considered Motion and Magnitude by themselves in the abstract. The fourth and last part, concerning the Phaenomena of Nature, that is to say, concerning the Motions and Magnitudes of the Bodies which are parts of the World, reall and existent, is that which followes.