Mathesis enucleata, or, The elements of the mathematicks by J. Christ. Sturmius ; made English by J.R. and R.S.S.

CHAP. I. Containing the Definitions or Explications of the Terms which relate to the Object of Mathematicks.

DEFINITION I.

MAthematicks is the Science or Knowledge of Quan∣tity, and of Beings, as far as they are subject to it, or measurable; and may justly claim the Name of Ʋniversal, while it is employ'd in Demonstra∣ting those Properties which are common to all or most Quan∣tities: But when it descends to the different Species of Quantity, and is busied in contemplating the Affections belonging par∣ticularly to this or that Quantity, it is distinguished by va∣rious Names, and distributed into various Parts, according to the various diversities of the Objects.

DEFINITION II.

QUantity may be defined in General, whatever is capable of any sort of Estimate or Mensuration, as immediately the Habitudes and Qualities of Things, as e. g. the multitude of Stars in the Heaven, or of Souldiers in an Army, the length of a Rope, or Way, the weight of a Stone, the swiftness or slowness of Motion, the Price of Commodities, &c. but medi∣ately, the very things themselves wherein those Estimable Qua∣lities are inherent. Whence with the ingenious Weigelius we may not incongruously reduce them all to these four Kinds or Genders, viz. 1. to Quanta Naturalia, Natural Quantities, or such as Nature has furnish'd us with, as Matter with its Exten∣sion and Parts, the Powers and Forces of Natural Bodies, as Gravity, Motion, Place, Light, Opacity, Perspicuity, Heat, Cold, &c. 2. to Moral Quanta or Quantities, depending for the most part on the Manners of Men, and arbitrarious De∣terminations of the Will; as for Example, the Values and Price of Things, the Dignity and Power of Persons, the Good or Evil of Actions, Merits and Demerits, Rewards and Punish∣ments, &c. 3. to Quanta Notionalia, arising from the Notions and Operations of the Understanding, as e. g. the amplitude or narrowness of our Conceptions, universality or particularity, &c. in Logick; the length or brevity of Syllables, Accent, Tone, &c. in Grammar: And lastly, to Quanta Transcendentia, Tran∣scendent Quantities, such as are obvious in Moral, Notional, and Natural Beings; as Duration, i. e. the Continuation of the Existence of any Being; which in Physicks especially is named Time, and may be conceived as a Line, &c. To these you may moreover add Ʋnity, Multitude or Number, Necessity and Contingency.

DEFINITION III.

NƲmber (whereon we shall make some special Remarks) if it be taken in the Concrete, is nothing else than an Aggregate or Multitude of any sort of Beings; taken abstract∣edly, it is, as Euclid calls it, 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉, a multitude, or (as they call it) Quotity of Ʋnities, on the one hand Number, i. e. many are opposed to one; and in that sense Unity is not a Num∣ber:

On the other hand Unity may be esteem'd a Number, since it is no less (if I may be allow'd that term) some Quo∣tity than two or three. But as we denote or signifie particular Things, when we speak of them Universally, by the Letters of the Alphabet, A, B, C, (a, b, c,) &c. as universal Signs or Symbols of them; so for distinctly and compendiously Expres∣sing the innumerable Variety of Numbers, Men have found out various Notes, the most natural whereof, are Points disposed in particular extended Orders, as . . . to denote Three,

[illustration] 9 dots arranged in 3 rows and 3 columns

to denote Nine, &c. But that way which is most commodious for Practise, is by the common Notation, or Cyphers, 1, 2, 3, 4, 5, 6, 7, 8, 9. the invention whereof, as we have it by Vulgar Tradition, is owing to the Arabians. By a very few of these we express any number tho never so great, by a won∣derful, tho now adays familiar, Artifice; the first Inventor of them having Establish'd this as an arbitrary Law, that the first of them shall signifie unity or one; the 2d two, &c. as often as they stand alone; but placed in a row with others, or on the left hand of one or more 0, or noughts, (which of themselves stand for nothing, but fill up empty places) if in the second, be∣fore a nought, they denote Tens; if in the 3d. Hundreds; in the 4th Thousands; in the 5th Myriads or Tens of Thousands; in the 7th so many Thousands of Thousands, or Millions; in the 8th Tens of Millions, &c. and so onwards, increasing al∣ways in decuple Proportion, by Tens, Hundreds, Thousands, &c.

COROLLARY I.

HEnce you have a way of expressing or writing any Sum by these Notes, which you may hear expressed in Words; as if we were to express in Notes the year of our Lord, One Thousand Six Hundred Ninety and Nine, it is manifest, that according to the method above described, by placing 9 on the right hand in the first place, and nine again in the second to∣wards the left, six in the third, and one or unity in the fourth, the business will be done. Thus it will be easie to any one with a little attention, to express any Number whatsoever by these Notes; (as suppose that which Swenterus proposes, in Delic-Physico-Math.

Part. 1. Probl. 75.) Eleven Thousand, Eleven Hundred, and Eleven.

COROLLARY II.

HEnce you have also the Foundation and Reason of the Rule of Numeration in Arithmetick, or expressing any Numbers or Sum in Words, which you see written in Cyphers; which for greater ease may be done thus, viz. beginning from the first Figure towards the right hand, over every fourth Figure note a Point, (including always that which was last pointed) and at every second Punctation or Point, draw short strokes thus, one over the 2d, two over the fourth, &c. the first denoting Millions, the second Millions of Millions, or Bimillions, the third Trimillions, &c. and the Intercepted Points the Thousands, in their kinds, &c.

SCHOLION.

HEre I cannot omit, on this occasion, what the foremen∣ti ned Weigelius has hinted about another way of Nume∣ration, and which Dr. Wallis mentions, Oper Mathemat. Part 1. p. 25. & 66. shewing there a way (and illustrating it by Ex∣amples) of Numeration, and of Expressing the Figures; which proceeds thus; whereas now adays in numbring we ascend from uni y or 1 to ten (the reason whereof; after which Aristotle makes a prolix Inquiry, Probl. 3. Sect. 15. was taken without doubt from the denary Number of our Fingers) if from unity we proceed only to four, (which Aristotle in the same place tells us some of the Thracians used to do of old,) and thence retur∣ning back again to Unity, we should proceed again after the same way; we might after that way obtain a vastly more simple and easie Arithmetick, than we have now adays. Which, even hence we may conclude; because for Multiplication and Divi∣sion there would need no other Table (or Pythagorick Abacus) than this easie and short one:

• 1.1.1 once one is one;
• 2.2.10 i. e. twice two are four;
• 2.3.12 twice three are four and two.
• 3.3.21 thrice three are twice four and one

[illustration]

Pag. 5.

Fig. 1.

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And altho it is pity that we can't hope now a-days to substi∣stute this vastly easier way of Computing, in room of the other now in use, because the other is universally receiv'd, and most sorts of Measures and other Quantities are fitted and accommo∣dated to the decuple Proportion; yet it ought not to be alto∣gether neglected in Mathematicks, which might receive very great advantage hereby, especially in Trigonometry, if the in∣genious Invention of the Logarithms had not already supply'd its use therein. The whole Foundation of this Tetractys or Quaternary Arithmetick, is placed only in these three Notes, 1, 2, 3; ••o that any one of them alone, or in the first place, should denote Units, in the second place, Tetrads, or so many Fours (or Quaternions,) in the third place so many Sixteens, in the fourth place so many times Four Sixteens, or 64.s. &c. al∣ways proceeding in a Quadruple Proportion. For which way of Numeration there might be found out terms as commodious as those we now use, and which are thereby grown Familiar to us, as one, ten, twenty, a hundred, a thousand, &c. which will be evident by what follows:

One, Ʋnum 1

One.

Ten, Decem 10

Quatuor, Tetras, a Quaternion or Four.

Twenty, Viginti 20

a Biquaternion

Thirty, Triginta 30

a Triquaternion

Hundred, Centum 100

a Tetraquaternion

Thousand, Mille 1000

a Quartan

Ten Thousand, 10000 &c.

a Tetraquartan, &c.

DEFINITION IV.

A Magnitude is whatever is conceived to be Extended or Continuous, or has parts one without another, and con∣tained within some common Term or Terms: wherein that is called a Point which is conceived (as indivisible, or) to have no Parts, and so no Magnitude, but is notwithstanding the be∣ginning or first Principle of all Magnitude.

DEFINITION V.

IF we conceive a Point (A) (Fig. 1.) to be moved towards B, by this motion it will leave a trace, or describe the Magni∣tude

AB of one only Dimension, that is Length without Lati∣tude, or which at least we are to conceive so, and is called a Line: If that Line AB be conceiv'd again so to be moved, as that its extreme Points AB shall describe other Lines BC and AD, it will describe by that Motion the Magnitude AB CD, or (to denote it more compendiously by the Diagonal Letters) AC or BD, having both length and breadth, but without any depth or thickness, or at least so to be conceived, and this is called Superficies or Surface: Lastly, if this Surface AC be con∣ceived so to move, e. g. upwards or downwards, that its opposite Points A and C again describe other Lines AF and CH, and consequently each of its Lines other Surfaces, &c. by this Mo∣tion there will be formed a Magnitude of three Dimensions, which we call a Solid or Body, which we will also denote by the two diametrically opposite Letters AH and DG. But as this Motion of the Point, Line, or Surface, may be various, so there will be produced by them various sorts of Lines, Sur∣faces, and Solids: But these Productions stop here, and proceed no further; for the Motion of a Body can only produce ano∣ther Body greater than the first, but no more new Dimensions.

CONSECTARYS.

I. POints therefore being moved thro' equal Intervals in the same or a like way or trace (e. g. in a streight or the shortest trace) describe equal Lines; and

II. The same or equal Lines moved thro' the same Right-lined or Curvilinear Paths, describe equal Surfaces; and

III. Equal Surfaces moved according to the same Methods and Conditions describe equal Solids: which, if rightly under∣stood, are the first certain and infallible Foundations of the Method of Indivisibles. But here you must take care to distin∣guish between the way which the Line it self describes, and that which its Ends or extreme Points describe: For altho e. g. the Point a (Fig. 24.) moves along in a more oblique way than A, and so describes a longer Line a c; yet the Line a b describes by a parallel Motion, an equal space with the Line A B, (viz.) the same which the whole Line A b, whereof they are parts, would describe. See Faber's Synopsis, p. m. 13.

DEFINITION VI.

BUT that we may a little further prosecute this Genesis of Magnitudes (as very much conducing to understand their Nature and Properties) if the Point A moves to B the shortest way, it describes the Right Line AB; but if in any (one) other it will describe the Curve or Compounded Line ACB: From whence, with F. Morgues, we may infer these

CONSECTARYS.

I. THat two Right Lines(a) 1.1 beginning from the same Point A, and ending in the same Point B, will necessarily coincide, nor can they comprehend or inclose Space; for if they did, one must deviate, and so would cease to be a Right Line.

II. In a Space comprehended by three Right Lines AB, BC, CA,(a) 1.2 any two taken together, must needs be greater than any one alone. Moreover we may add this before hand;

III. In a Circle a Right Line drawn from A to B (Fig. 3) will fall within the Circle, because the Curve Line ADB de∣scribed, as we shall hereafter shew, being longer than a Right Line, must necessarily fall beyond it, or on the outside of it. And lastly,

IV. A Tangent, or Line (b) which does not cut or enter into the Circle, touches it only in one Point.

Moreover if a Right Line AB (Fig. 4.) move on another Right Line BC, remaining in the same Position to it, it will generate a Plan Surface, to which a Right Line being any way applied, will touch it with all its point, as Faber rightly describes it; if a Right Line be moved on a Curve, or a Curve on a Right Line, &c. they will generate a Curve Surface, call'd Gib∣bous, or Convex without, and Concave within.

DEFINITION VII.

IF a Right Line be fixed at one of its ends A, and the other be moved round (Fig. 5.) it will describe in this Motion a

Circular Plane, or a Circle; and by the motion of its end or extreme Point B,(a) 1.3 the Periphery or Circumference of that Circle BEF.(b) 1.4 The fixed Point A is called the Center of that Circle; the Lines AB, AC, &c. its Radii or Semi-Diame∣ters; all of which are equal one to another. Any Right Line BC drawn from one part of the Circumference thro' the Center to another, is called the Diameter, and divides the Circle into two Semicircles BECB and BFCB. The Circumference of a Circle, whether great or small, is divided into 360 equal parts called Degrees, and each Degree into 60 Minutes, &c. From this Geniture of the Circle presupposed, there evidently fol∣low these

CONSECTARYS.

I. THat 2 Circles which cut one another cannot have the same common Center; for if they had, the Radii ED and EA drawn from the common Center E (Fig. 6.) would be equal to the common Radius EB that is the part to the whole.

II. Nor can two Circles touching one another within side, have one and the same Center, for the same reason.

III. Of Lines falling from any given Point without the (Fig. 7.) Circle,(b) 1.5 and(c) 1.6 passing thro' the Pe∣riphery to the opposite Concave part of it, that which passes through the Center of it, is the longest, viz. AB; and of the other that which is nearest to it is longer than that which is more remote: But on the contrary of those which fall on the Convex Periphery, that which tends towards the Center, as Ab, is the least, and the rest gradually greater, and there can be but two, as AE and AF, or Ae and Af, equal: All which will appear very evi∣dent by drawing other Circles from the Center A thro' B, D, E, and b, d, e. Or thus; having drawn two other Circles, from the Radii AB and Ab, if we conceive the Radii Ab and Cb to move towards the right hand, their ends will always recede further from one another; the same is also evident of the Ra∣dii AB and CB, moved also to the right together.

IV. Moreover (Fig. 8.) of all the Lines drawn within the Circle(a) 1.7 the Diameter is the greatest, and the rest gradually less, by how much the more remote they are from the Center, &c.

Which will be very evident to any one who contemplates a Circle inscribed in a Square, as also the Genesis of Curvity it self; as also many other ways which I shall now omit; or to mention one more thus; because the two Radii CA and CB being mo∣ved, in order to meet together, necessarily approach nearer to one another in their extreme Points.

DEFINITION VIII.

THE Aperture or opening of two Lines (Fig. 9.) AB, AD, &c. that are both fixed at one end at A, and the other ends opened or removed farther and farther from one another, is called an Angle,(a) 1.8 and usually de∣noted by 3 Letters, D, A, B, (whereof that which denotes the Angular Point, always stands in the middle,) and measured by the Arch of a Circle BD, or a certain num∣ber of Degrees which it intercepts. The greatest Aperture of all BAC is when the 2 Legs of the Angle AB and AC make one Right Line, and is measured by a Semicircle, or 180 De∣grees. The mean or middle Aperture BAE or CAE, when one Leg EA is erected on the other AB or AC at Right An∣gles, so that it inclines neither one way nor the other, (thence called a Perpendicular) is named a Right Angle, whose measure is consequently a Quadrant (or quarter part) of a Circle or 90 Gr. Wherefore a Semicircle is the measure of two Right An∣gles: An Aperture or Angle BAD less than a Right Angle (and so measured by less than 90 degrees) is called an Acute Angle; and that which is greater than a Right Angle, as DAC (and so consisting of more than 90 degrees) is called an Obtuse An∣gle. Whence me may now draw these

COROLLARYS.

I. TWO or more Contiguous Angles(a) 1.9 con∣stituted on the same Right Line BC, and at the same Point A (as DAB, and DAC or DAB, DAE and EAC) make two Right An∣gles, as filling the Semicircle; and consequently,

II. All the Angles that can be constituted about the Point A (as filling the whole Circle) are equal to 4 Right ones: As

also on the other side(a) 1.10 if two Right Lines AB and AD meet on the same point A of another Right Line AC, and make the Contiguous Angles equal to 2 Right ones, that is, if they fill a Semicircle, BC will necessarily be the Diameter of a Circle, and consequently a Right line.

III. If one of the Contiguous Angles BAE be a Right one, the other CAE will be so also.

IV. If two Right Lines AB, CD, cut one another in E, the 4 Angles they make will be equal to 4 Right ones.

V. And as it is evident at first sight (Fig. 10.) that any Circle having one half (or Semicircle) folded on the other, at the Diameter ECD, the two Semicircles EHD, and EID, must needs agree, or every where coincide one with the other; so if the Angle ACD be supposed equal to the Angle BCD, that is, the Arch AD to the Arch BD, having one Leg CK or CL common; the others AC and BC being supposed before equal,

1. The Bases BL and AL, KB and KA, will be also equal; for these will coincide too, and therefore the Angles also.

2. The Line AB being bisected in K, the two Angles(b) 1.11 (c) 1.12 at K will also coincide and be equal, and con∣sequently Right Angles: and contrarywise,

3. The Angles at the Base of equal(d) 1.13 Legs, CAB, CBA, and also those below the Legs, the Legs being produced to F and G, are equal.

4. Consequently the Spaces ACL and BCL, ACK and BCK are equal to one another.

5. The Contiguous Angles AED and BED insisting on equal Arches AD and BD are equal, and è contra; as also those that are not Contiguous, if their Vertex's are equidistant from E, &c.

6. It is hence also manifest, that a Perpendicular erected on the middle of any Line AB, inscribed in any Circle, passes through its Center, by what we have just now said; and if you likewise erect Perpendiculars on the middle of any 2 Lines, ab and bm (Fig. 11.) connecting any 2 Arches, or any 3 Points, a, b, m, that are not placed all in the same Right Line, those 2 Per∣pendiculars ke, no, will determine (by their Intersection) the Center of a Circle that shall pass through these 3 Points.

DEFINITION IX.

IF one Right Line DE cut or pass thro' another AB (Fig. 12) the opposite Angles at the top or intersection ACD and ECB are called Vertical; as also the other two ACE and DCB: Whence follow these

COROLLARYS.

I. THat the Vertical Angles are always(a) 1.14 Equal; for both ACD and ECB with the third, ACE, which is common to both, fill or are equal to a Semicircle; as likewise both ACE and DCB with the third ECB, which is common.

II. Contrarywise, if at(a) 1.15 the Point C of the Right Line DE, the 2 opposite Lines AC and CB make the Vertical An∣gles x and z equal, then will AC and CB make one Right Line; for, since x and o make a Semicircle, and z and x are equal, by Hypoth. o and z will also make or fill a Semicircle, whose Diameter will be ACB.

III. By the same Argument it will appear, that of 4 Lines(b) 1.16 proceeding from the same Point so as to make the opposite Vertical Angles equal, the 2 oppo∣site ones AC and CB, as also DC and CE, will make each but one Right Line; for since all the 4 Angles together make a whole Circle, or 4 Right Angles, and the sum of x and o is equal (by Hypoth.) to the sum of o and z, it follows, that both the one and the other will make Se∣micircles, whose Diameter will be AB and DE, and so Right Lines.

DEFINITION X.

IN any Circle, a Right Line, as D.G, that subtends any Arch of it DGB, is called the Chord of that Arch (Fig. 13.) BF (a part cut off from the Semidiameter BC passing thro' the middle of the Chord) is called the Sagitta or Intercepted Ax, but most commonly the Versed Sine; and DF let fall from the other ex∣tremity of the given Arch BD, on the Semidiameter at Right Angles, is called the Right Sine of that Arch BD, or of the Angle

BCD; also DI is called the Right Sine of the Complement (or for brevity sake, Sine Compl.) of that Arch DH, or Angle DCH, &c. but the greatest of all the Right Sines HC let fall from the other extremity (or end) of the Quadrant (which is indeed the same as the Semidiameter of the Circle) is called the whole Sine or Radius; lastly, BE is called the Tangent of the Arch BD or Angle BCD, and CE its Secant: Whence Mathe∣maticians, for the sake of Trigonometrical Calculations, have divided the whole Sine or Radius of the Circle into 1000, 10000, 100000, 1000 000, 10000 0000, &c. parts, thence to make a proportionable Estimate of the number of Parts in the Sine, Tangent, or Secant of any Arch, &c. as may be seen in the Tables of Sines, Tangents, and Secants. From these Suppositions and Explications of the Terms, we shall now in∣fer from this Definition the following

COROLLARYS.

I. IN equal Circles (and so much more(a) 1.17 in one and the same) as the Radii or Semidia∣meters BC and bc are equal, so also it is evident, that the Right Sines DF & Df, of equal Arches BD and bd, or equal Angles BCD and bcd, also the Tangents BE and be, and Secants CE and ce, and Subtenses or Chords DG and dg, also the Sagittae or intercepted Axes BF and bf, of double the Arches DBG and dbg, &c. will be equal, and so consist of an equal number of Parts of the whole Sine or Radius, &c. which both is evident from what we have said before, and may be further evinced, if one Circle be con∣ceived to be put on the other, and the Radius BC on the Ra∣dius bc, that so they may coincide, by reason of the equality of the Arches BD and bd; and so of all the rest. And è contra,

II. In unequal Circles, the Sines, Tangents, &c. of equal Angles BCD or bcd (Fig. 14.) or similar Arches, or Arches of an equal number of Degrees, BD and bd, will be also simi∣lar or like, &c. i. e. the Sine df consists of as many parts of its Radius bC, as the Sine DF does of its Radius BC, &c. e. g. if the Radius BC be double of the Radius bc, each thousandth part of the one, will be double of each thou∣sandth

[illustration]

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part of the other, but they are alike 1000 in each; be∣cause the degrees in the Circumference of the little one, parti∣cularly in the Arch bd are but half as big as those in the Arch BD, and yet equal in number in both. Thus also if the Sine DF contains 700 of the 1000 parts of its Radius BC, df will also contain 700 of the 1000 parts of its smaller Radius bc, and in like manner the Chords DG and dg, and the Tan∣gents BE and be, &c. contain a like number of parts, each of its own Radius.

SCHOLION.

IT may not be amiss here to note by the by (altho it may seem more proper to be taught after the Doctrin of Pro∣portions) that if, v. g. the degrees of a greater Circle be each of them respectively double, or triple, or quadruple, &c. of the degrees of a less Circle, according as the Radius of the one is double or triple to the Radius of the other, then, at least as far as Mechanical Practice can require, you may find the Arch of a greater Circle equal to the whole Periphery of a less, viz. if you take reciprocally that part of the greater Periphery, which shall be as the Radius of the less to the Radius of the greater, or as one degree of the less Periphery to one degree of the greater. e. g. if the less Radius bc be half the greater BC, and so also the Periphery, and each of the degrees of the one, be one half of the Periphery, and of each of the degrees of the other, one half of the greater Periphery will reciprocally be equal to the whole less Periphery, or 180 degrees of the one to 360 of the other, &c.

2. The same (at least in this case where the Radius cb is double of the Radius CB) may be done also Geometrically by the same reason. Having described Circles on each Radius, suppose the Radius CB (Fig. 15.) so to move with an equable motion about its Center c, as to take or move the Radius of the greater Circle cb along with it, and coming, v. g. to I. stops that also at 1, and going forward, to II. stops that again at 2, &c. Hence it will be manifest to any attentive Reader, that when the less Radius CB shall have described the Semicircum∣ference B. II. III. the greater Radius cb having moved to 3, will have described precisely a quarter of its circumference;

and if still the less Radius C. IV. moves on to the right hand, and continues to carry the greater c 4 along with it onwards the same way, it will necessarily follow, that in the same mo∣ment as the Radius C. IV. (together with c 4.) shall come to its first situation in B, having described a whole Circle, the op∣posite Radius c 4 will be come to 5, and have described half its Circle, having moved all along with an equal Motion. Hence it is evident, that the whole least Circle answers exact∣ly to half the greater, and half of the first to a quarter of the last; as also the Quadrant B. II. to the Octant (or 8th. part) b 2, &c. whence any Arch being given, as B. I. in the least Circle, if you draw thro' I the Radius of the greater Circle c 1. you'l cut off an Arch b 1. equal to the given Arch in mag∣nitude, but only half in the number of degrees.

3. Hence follows naturally that celebrated Proposition of Euclid, that the Angle at the Center BC. I. or BC. II. is double of the corresponding Angle at the Periphery bc. 1. or bc. 2, &c. which in this case is manifest, and in the other 2 (Fig. 16.) of the wholes or remainders DCD and DPD it is also(a) 1.18 certain; which is true also of the parts BCD and BPD to be added or subtracted by the first Case.

(b) 1.194. Hence we have a new way of bisecting any given Angle CDE, or Arch CE (Fig. 17.) viz. if you make CB equal to the Leg DC, and from this, as Radius, describe an Arch BF equal to the Arch CE, and draw DF.

And with the same facility we might obtain the Tri∣section, if the greater Radius being triple to the less, was thus carried along by an equable Motion, as we have shewn how to do already in a double Radius; and this at first sight may seem very probable.

But whether the triple Radius be immediately carried round by the simple Radius CB, or by means of the double Radius cb, neither the one nor the other will cause an equable Motion. For in the latter Case, while the Radius cb describes the quadrant Bb, the Radius de will not describe so much as a Quadrant; but while cb with the same velocity describes the other Quadrant bf, the Radius de will come to g, de∣scribing an Arch as much greater than a Quadrant as the for∣mer

was less. In the former case on the contrary, the Radius CB moved on to D beyond a Quadrant, while de was carried from B to e, but if the Radius CD moving on, should again carry de along with it, the one would describe the same Arch eB, while the other would describe one less than before.

5. Hence the Angle at the Center ACE (F g. 19.) upon the Arch AE, is equal to the Angle ADB at the Periphery, upon double that Arch AB.

6. Hence the Angle ADB in the Semicircle (Num. 1.)(a) 1.20 is a Right Angle, in a Segment less than a Semicircle (Num. 2.) is an obtuse Angle, and in a greater (Num. 3) an Acute one, because the Angle at the Center ACE upon the half Arch, is equal to the Angle ADB pr. praeced. 5. and is a Right Angle in the first Case, Obtuse in the second, and Acute in the third.

7. Hence Angles in the same Segment, or(b) 1.21 on equal Segments of equal Circles, or on the same or equal Arches, are all equal and è contra.

DEFINITION XI.

WHen 2 or more Lines AB and CD are so continued as to keep always the same distance from one another (whose Genesis may be conceived to proceed from the uniform Motion of 2 Points A and C, always keeping the same distance from each other) they are called Parallels: But as it evidently fol∣lows from this Definition, that(a) 1.22 those Lines which are Pa∣rallel to one third, are parallel to one another (since adding or subtracting equal Intervals to or from other equal ones, the sums or remainders must needs be equal;) so if the Parallels are Right Lines and cut transversly (or slopingly a-cross) by another Right Line EF, you'l have these

COROLLARYS.

I. THE Angles(b) 1.23 which we call Alternate ones, GHK and HGI (Fig. 21.) are equal by Corollary I. Definition X. since the distances GK and HI, which are the Right Sines of the said Angles, are supposed equal.

II. The External Angle EGA is also equal to the Internal opposite Angle GHK, by Consect. 1. Definit. 9. because that External Angle EGA is equal to the alternate Vertical one HGI.

III. The same Internal Angle GHK, with the other internal opposite one on the same side AGH (as well as the External one EGA, by Coroll. 1. Definit. 8.) are equal to two Right ones.

IV. On the contrary, If any Right Line EF(a) 1.24 cutting 2 others AB and CD transversly, makes the alternate Angles GHK and HGI equal, their Right Sines, by Consect. 1. Definit. 10. will be equal, and consequently the Lines AB and CD parallel: and the same will follow, if the External Angle be supposed equal to the Internal, or the 2 Internal ones on the same side equal to 2 Right ones; since from either Hypothesis the former will immediately follow.

V. From whence it appears more than one way (b) That the 3 Internal Angles of any Triangle (e. g. H, G, K, which will serve for all) taken together,(a) 1.25 are equal to two Right ones, and the External one GHD is equal to the two Internal opposite ones. For we might either conclude with Euclid, that 1, 2, 3, together make 2 Right ones, by Consect. 1. Definit. 8. but 2=II and 3=III pr. 1 and 2 of this, therefore I, II, III=2 Right ones; or with others, 1, II, 4 are = 2 R. but 1=I and 4=III pr. 1st of this. There∣fore, &c. or more briefly with F. Pardies, 1=I pr. 1st of this, but 1, II, III, together = to 2 Right ones, by the 3d of this; therefore I, II, III=2 R. Q. E D.

DEFINITION XII.

IF a Right Line AB (Fig. 22.) be conceived to move from the top of a plain Angle CAD with a motion always paral∣lel to its self, so that at one end A it shall always touch the Leg AC, and all along cut the Leg AD, while at length being come to F, it shall only touch that Leg with its other end B, and so fall at length wholly within the Angle CAD: It will describe by this motion within the Legs CAD the Triangular Figure EAF, and without them the Triangular Figure BAF; its parts within them a f continually increasing, and the other without

fb continually decreasing; but with all its Parts, or the whole Line, it will describe the Quadrangular Figure AEFB: Conse∣quently if the other Leg AD of the given Angle CAD (Fig. 23.) or any part of it AB, be moved along the other Leg remain∣ing parallel to it self, it will also describe a Quadrilateral Fi∣gure, which will be also equilateral, if the Line describing it AB, be equal to the Line AE according to which it is directed; but if either of the Lines, as AD be greater than the other, the opposite Sides will be only equal; for the describent or describ∣ing Line is always necessarily equal to its self, and the Points A, B, D, moved with an equal Motion, describe also in the same time equal Lines AE, BF, DG. From these Geneses of Quadrangles and Triangles we have the following

CONSECTARIES.

I. THese Quadrilateral Figures are also Parallelograms, i. e. they have their opposite Sides Parallel;(a) 1.26 because the Line that describes them is supposed to re∣main always parallel to its self, and the Points A and D, or A and B, to be always equidistant.

II. Because the 2 Internal opposite Angles(b) 1.27 A and E, and also E and F, &c. are equal to 2 Right ones, by Consect. 3. Definit. 11. if one Angle v. g. that at A be a Right one, all the others must neces∣sarily be so too [in which case the quadrilateral and equilateral Figure AF is called a Square, and the other AG an Oblong, or Rectangle:] if there be no Right Angle, the opposite Angles transversly or cross-ways, a and f, or a and g are equal, because both the one and the other, with the third (e) make 2 Right ones [in which case the quadrilateral Equi∣lateral af is called a Rhombus, but the other ag a Rhomboid.

III. The Transversal (or Diagonal) Line(c) 1.28 in any Parallelogram, divides it into two equal Triangles AEF and FAB; for all the Lines and Angles on each side are equal, and as the descri∣bent (Line) AB moved thro' the Angle EAF upon the Line AE described the Triangle AEF; so the Line EF, equal to the former, moved after the same way, thro' the Angle AFB also equal to the former Angle, upon the equal Line FB, must

necessarily describe an equal Triangle; or, in short, all the In∣divisibles af, or their whole increasing Series, are necessarily equal to the like number of Indivisibles fb, increasing recipro∣cally after the same way.

IV. All Parallelograms that are between the same Parallels AB and CF (Fig. 24.) i. e. having the same Altitude(a) 1.29 and the same or equal Bases, as CD or CD and cd, are equal among themselves; for they may be conceived to be described by the equal Lines AB and ab equally moved thro' the same or equal Intervals of the Parallel Lines; so that all or each of the Indivisibles or Elements AB will necessarily be equal to all and each of the Indivisibles ab; for they all along answer one to the other both in num∣ber and magnitude.

SCHOLIUM.

HEre you have a Specimen of the Method of Indivisibles, introduced first by Bonaventura Cavallerius, and since much facilitated; and altho these Indivisibles placed one by another, or as it were laid upon an heap, cannot compose any Mag∣nitude, yet by an imaginary Motion they may measure it, and as it were, after a negative way, demonstrate the Equality of two Magnitudes compared together, viz. if we conceive a cer∣tain number of such Elements in any given Magnitude, and thence conclude that in another consisting of the like Elements, ordered or ranked after the same way, there can be neither more nor less in number than in the first; thence follows their Equality, &c.

V. Hence therefore it is also Evident, that Triangles upon the same and equal Bases as CD and cd, and placed between the same Parallels, are necessarily equal, because they are the half of equal Parallelograms AD and ad, by the 3d Consectary of this Definition.

VI. F. Mourgues ingeniously concludes from hence, viz. be∣cause the 2 Internal opposite Angles(a) 1.30 (b) 1.31 on the same side in any Parallelogram, are equal to two Right ones, and so all together equal to four; that therefore the three Angles of any Triangle ABC (Fig. 25. which may always be compleated into a

[illustration]

Pag. 18.

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Parallelogram) are equal to half of those four, or to two Right ones. This may be yet more briefly conceived thus; the Angles b+c+d (the sum of the two inferiour ones) = to two Right ones; but a = alternate d: therefore b+c+a = to two Right ones, Q. E. D.

VII. Because it is manifest in Rectangular Parallelograms, if the Altitude AB, and Base BC (Fig. 26.) measured and divided by the same common Measure, be conceived to be (multiplied or) drawn one cross the other, that the* 1.32 Area AC, thereby described, will be divided into as many little square Measures or Area's, as the number of their Sides multiplied together would produce Units; therefore the A∣rea of any other Parallelogram will be after the like manner produced, if the Base be multiplied by the Perpendicular heigth, equally as if it were a Rectangle of the same Base and Alti∣tude.

VIII. Consequently also you may have the Area of any Triangle, by Consectary 3 and 5. if the Base be multiplied by half the Perpendicular heigth; or, the whole Base being multiplied by the heighth, if you take the half of the product.

DEFINITION XIII.

BUT as there are various Species of Triangles, while first with relation to their Sides, one is called Equilateral, as ABC (Fig. 27.) because all its Sides are equal; another Equi∣crural or Isosceles, as DEF, because it has two equal Sides DE and EF, while its Base DF may be either longer or shorter; and a third is called Scalenum, as GHI, because it has all its Sides un∣equal; then again in respect to their Angles, one is called Rectangled, as a, b, c, because it has one Right Angle at a;

Another Obtusangled, as d, e, f, because it has one obtuse An∣gle at d; a third is called Acuteangled, as g, h, i, because all its three Angles are Acute: So each of these kinds has its pecu∣liar properties, which we shall partly hereafter demonstrate in their proper places, and partly deduce here as

CONSECTARYS.

I. ALL Equilateral Triangles are also Equiangular, and con∣sequently Acutangled; for, having found a Center for three Points, and the Periphery A, B, C, (see Fig. 28.) by Consect. 6. Definit. 8. the three Arches AB, BC, and AC, answering to equal Chords; and consequently the three Angles at the Center O are equal, by Consect. 1. of Definit. 10. and therefore the three Angles at the Periphery also, as being half of the other, by the 3d Consectary of the same Definition. Each Angle therefore is one third part of two Right ones, by Consect. 6. Definit. 12. two thirds of one Right Angle, i. e. 60 degrees, and consequently Acute.

II. It follows also by the same Reason in an Isosceles Triangle, that the Angles at the Base opposed to equal Sides are equal, and(a) 1.33 consequently Acute; for having circum∣scribed a Circle about it, equal Arches will cor∣respond to the equal Chords DE and EF, and equal Angles at the Center DOE and FOE will correspond to them, and equal ones at the Peri∣phery DFE, and FDE to these again. And it is evident that each of these are less than a Right Angle h. e. an Acute one, because all three are equal to two Right ones. Wherefore if the third is a Right Angle, the other two at the Base will ne∣cessarily be half Right ones.

SCHOLIUM.

WE will here (a) shew by way of Anticipation, the truth of the Pythagorick Theorem, esteemed worth an He∣catomb: Which hereafter we will demonstrate after other dif∣ferent ways; viz. In a Right Angled Triangle BAC (Fig. 29.) the Square of the greatest Side opposite to the Right Angle, is equal to the Squares of the other two Sides taken together. For having de∣scribed the Squares of the other two Sides, AC dE, DE ab (taking ED=AB) and the Square of the greatest BC cb, it will be evident, that the parts X and Z are common to each, and that the two other Triangles in the greatest Square BAC and BDb, are equal to the two Triangles bac and Cdc which

remain in the two Squares of the lesser Sides; and so the whole truth of the Proposition will be evident, while these two things are undoubtedly true: 1. That the Side of the greatest Square Bb will necessarily concur with the Extremity of the less Db, and the other Side of the greatest Square Cc with its Extremity c, will precisely touch the Continuation of the Sides of the two least Squares dEa; as you'l see them both expressed in the Fi∣gure. 2. The said two Triangles are every way equal; for the Angles at C with the intermediate one at Z, make two Right ones, therefore they are equal; but the Side CA is equal to the Side cd, and CB to Cc, and the Angles at A and d Right ones. Wherefore if we conceive the Triangle ABC to to be turned about C, as a Center to the right hand, it will exactly agree with the Triangle Cdc, and the Point B will ne∣cessarily fall on the continued Line d E, as agreeing with the Line AB. Hence it is now evident, that Ca=BD, and be∣cause ba is also = bD, and the Angles at a and D Right ones. Where, if we conceive the Triangle bac to be moved about b as a Center, untill ba coincides with bD, and ac with DB, bc will also necessarily coincide with bB Q. E. D.

To this Demonstration of Van Schooten's, which we have thus illustrated and abbreviated, we will add another of our own, more like Euclids, but somewhat easier, which is this: Having drawn the Lines (as the other Figure 29 directs) the ▵ ACD being on the same Base AC with the Square AI, and between the same Parallels, is necessarily one half of it, but it is also half of the Parallelogram CF being on the same Base with it, viz. DC; therefore this Parallelogram = ▭ AI. In like manner ▵ ABE is half the ▭ AL, and also half the Parallelogram BF, therefore BF=▭AL: therefore CF + BF that is the ▭ of BD = to the two ▭ ▭ AI + AL. Q. E. D. For because the Side BE occurs to, or meets the Side LK, and the Side CD the Side IH continued, it yet more apparently follows; because the An∣gles a and b, and also c and d, are manifestly equal, as making both ways, with the Intermediate x or z, Right Angles. There∣fore the ▵ BAC being turned on the Center B and laid on BLE will exactly agree with it, and turned on the Center C and laid on CID, will agree with that also, &c.

DEFINITION XIV.

ALL Rectilinear or Right Lined Figures that have more than three or four Sides (to the latter sort of which, there remains to be added another Species besides Pa∣rallelograms, call'd Trapeziums, whose Angles and Sides are un∣equal, as K, L, M, N, Fig. 30.) are called by one common Name Poligons, or Many-sided and Many-Angled Figures, and par∣ticularly according to the Number of their Sides and Angles, Pen∣tagons, Hexagons, Heptagons, &c. All whereof, as also Trapezia, being resolvible into Triangles by Diagonal Lines, (as may be seen in the 31 and foregoing Fig.) you have these

CONSECTARYS.

I. YOU have the Area of any Polygon by resolving it into Triangles, and then adding the Area's of each Tri∣angle found by Consect. of Definition 12 into one Sum.

II. The Area of the Trapezium KLMN (in the first of the Fig. 30) whose two opposite Sides, at least KL and MN, are Parallel, may be had more compendiously, if the Sum of the Sides be multiplied by half the common heigth KO.

SCHOLIUM.

HEnce we have the foundation of Epipedometry or Masuring of Figures that stand on the same Base, and Ichnography; in the Practise whereof this deserves to be taken special Notice of, that to work so much the more Compendiously, you ought to divide your Figure into Triangles, so that (Fig. 31.) 2 of their Perpendiculars may (as conveniently can be) fall on one and the same Base. For thus you'l have but one Base to measure, and 2 Perpendiculars to find the Area of both: But for Ichnography, the distance of the Perpendiculars from the nearest end of the Base must be taken; which we shall supersede in this Place and Discourse more largely on hereafter.

2. This resolution of a Polygon into Triangles may be perform'd by assuming a point any where about the middle, and making the sides of the Polygon the Bases of so many

Triangles; (see the 2d Figure mark'd 31) wherein it is evi∣dent; 1 That all the Angles of any Polygon are equal to twice so many right ones, excepting 4, as the Polygon has sides; for it will be resolv'd into as many Triangles as it has sides, and each of these has its Angles equal to 2 right ones. Subtract∣ing therefore all the Angles about the Point M (which always make 4 right ones by Cons. 2. Def. 8.) there remain the rest which make the Angles of the Polygon. 2 All the external Angles of any right lined Figure (e, e, e, &c.) are always equal to 4 right ones; for any one of them with its Contiguous internal Angle is equal to 2 right ones pr. Consect. 1 of the said Def. and so altogether equal to twice so many right ones as there are Sides or internal Angles of the Figure. But all the inter∣nal Ones make also twice so many right Ones, excepting 4 therefore the external Ones make those 4.

DEFINITIOF XV.

AMong all these plain Figures those are call'd Regular whose Angles and Sides are all equal, as among trilateral Figures the Equilateral Triangle, among Quadrilateral ones, the Square, and in other kinds, several Species which are not particulariz'd by Names; but all others in whose Angles or Sides there is any inequality, are call'd Irregular: Tho' some of these also, and all the other may be inscrib'd in a Circle. Whence you have these

CONSECTARYS.

I. THE Areas of the Regular Figures may be obtained yet easier, if having found their Center (by Consect. 6. De∣finit. 8.) you draw from thence the Right Lines CB, CA, &c. (Fig. 32.) till there be form'd as many Triangles ACB, ACF, &c. as the Figure has Sides; for since all these Triangles have their Bases AB, BF, as so many Chords, and their Altitudes CD, CG, as so many parts of intercepted Axes DE and GH, and also equal pr. Consect. 1. Definit. 10. and so by Consect. 5. Definit. 10. are equal among themselves; one of their Area's being found and multiplied by the number of Sides, or half the Altitude by the Sum of all the Sides, you'l have the Area of the

whole Polygon: For it is manifest from what we have already said, and very elegantly Demonstrated by F. Pardies, That any Regular Polygon inscribed in or circumscribed to a Circle, is equal to the Triangle Aza, one Legg whereof is equal to the Perpendicular heighth let fall from the Center upon any Side, and the other to the whole Pe∣riphery of the Polygon. Now if the Triangles into which the Poly∣gon is resolved, do all stand on the same Right Line Aa, (Fig. 32) and are all equal and of the same heighth, to which the Per∣pendicular AZ is equal, it will necessarily follow, that each pair of Triangles ABZ and ABC, BZF and BCF, &c. are equal among themselves, pr. Consect. 5. Definit. 12. and consequently the Sum of all the former will be equal to the Sum of all the latter, that is, the Triangle Aza to the Polygon given.

II. Since Regular Figures inscrib'd in a Circle, by bi∣secting their Arches AB, BF, &c. may be easily conceived to be changed into others of double the number of Sides, (as a Pentagon into a Decagon, &c.) and that ad Infinitum; a Circle may be justly esteemed a Polygon of infinite Sides, or consisting of an infinite Number of equal Triangles, whose common Al∣titude is the Semidiameter of the Circle: So that the Area of any Circle is equal to a Right Angled Triangle (as AZa) one of whose Sides AZ is equal to its(a) 1.34 Semidia∣meter, and the other Aa to its whole Circum∣ference.

SCHOLIUM.

IT may not be amiss to note these few things here, concer∣ning the Inscription of Regular Figures in a Circle.

I. Having described a Circle on any Semidiameter AC,(a) 1.35 (Fig. 33. N. 1.) that Semidiameter being placed in the Circumference, will precisely cut off one sixth part of it, and so become the Side of a Re∣gular Hexagon: and so the Triangle ABC will be an Equilateral one, and consequently the An∣gle ACB and the Arch AB 60 Degrees, by Cons. 1. Definit. 13.

II. Hence a Right Line AD, omitting one point of the di∣vision B, and drawn(b) 1.36 to the next D, gives you the Side of a Regular Triangle inscrib'd in the Circle, and subtends twice 60, i. e. 120 Degrees.

[illustration]

Pag. 24.

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40

III. If the Diameters of the Circle AD and DE (N. 2.) cut one another at Right Angles in the Center C, the Right Lines AB, BD, &c. will be the Sides of an inscribed Square ABDE: For(c) 1.37 the Sides AB, BD, &c. are the equal Chords of Quadrants, or Quadrantal Arches, and the Angles ABD, BDE, &c. will be all Right ones, as being Angles in a Semicircle (per Schol. 6. Definit. 10.) composed each of two half Right ones, by Consect. 2. Definit. 13.

IV. Euclid very ingeniously shews us how to Inscribe a Re∣gular Pentagon also, Lib. 4. Prop. 10 & 11. and also a Quin∣decagon (or Polygon of 15 Sides) Prop. 16. But though the first is too far fetch'd to be shewn here, yet (supposing that) the second will easily and briefly follow:

In a given Circle from the same point A (N. 3.) inscribe a Regular Pentagon AEFGHA, and also a Regular Triangle ABC; then will BF be the Side of the Quindecagon, or 15 Sided Figure. For the two Arches AE and EF make together 144 Degrees, and AB 120: (a) Therefore the difference BF will be 24, which is the 15th. part of the Circumference.

V. The Invention of Renaldinus would be very happy, if it could be rightly Demonstrated; (as he supposes it to be in his Book of the Circle) which gives an Universal Rule of dividing the Periphery of the Circle into any number of equal Parts re∣quired, in his 2d Book De Resol. & Comp. Mathem. p. 367. which in short is this: Upon the Diameter of a given Circle AB Fig. 34.) make an Equilateral Triangle ABD, and having divided the Diameter AB into as many equal Parts, as you design there shall be Sides of the Polygon to be Inscribed, and omitting two, e. g. from B to A, draw thro' the beginning of the third from D, a Right Line, to the opposite Concave Circumference, and thence another Right Line to the end of the Diameter B, which the two parts you omitted shall touch thus, e. g. for the Triangle, having divided AB into three equal parts, if omitting the two B2, thro' this beginning of the 3d you draw the Right Line DIII, and thence the Right Line III.B, which will be the Side of the Triangle; and so IV.B will be the Side of the Square, VB the Side of the Pentagon, &c.

N. B. The Demonstration of these (Renaldinus adds, p. 368.) we have several ways prosecuted in our Treatise of the Circle: Some of the most noted Antient Geometricians, have spent a great deal of pains in the Investigation and Effection of this Problem, and several of the Moderns have lost both time and pains therein: Whence, we hope, without the imputation of Vain Glory, we may have somewhat obliged Posterity in this point.

DEFINITION VI.

IF the Plane of any Parallelogram AC (Fig. 25.) be concei∣ved to move along a Right Line AE, or another Plane AF downwards, remaining always Parallel to its self; there will be generated after this way a Solid having six opposite Planes Pa∣rallel, two whereof, at least, will be equal to one another, whence it is called a Parallelepiped; and particularly a Cube or Hexaedrum, if the Parallelogram ABCD that describes it be a Square, and the Line along which it is moved, AE, equal to the Side of that Square, and Perpendicular to the describing Plane, and consequently all the six Parallel Planes comprehend∣ing this Solid, equal to one another. But if the describing or Plane Describent (Fig. 36.) be a Triangle or Polygon, the So∣lid is call'd a Prism, if a Circle, it is called a Cylinder. Now from the Genesis of these Solids you have the following

CONSECTARYS.

I. IF the Planes or Parallelograms Describent(a) 1.38 ABCD and abcd (Fig. 37.) are equal, and their Lines of Motion AE and ae also equal; the Solids thereby de∣scribed, viz. Parallelepipeds, Cylinders, and Prisms, (which will therefore have their Bases and heigths equal) will be equal among themselves; be∣cause the describent Indivisibles of the one, will exactly answer, both in number and position, to those of the other, as we have already shewn in Parallelograms; Consequently therefore,

II. Any Parallelepiped(b) 1.39 may be divided by a Diagonal Plane BDHF (or a Plane passing thro' its Diagonals) into two equal Prisms; for by

Consect. 3. Definit. 12. the Triangles ABD and BCD, are equal, and are supposed to be moved by an equal Motion thro' equal spaces.

III. And since it is evident, even by this Genesis of them, that in Right Angled Cubes and Parallelepipeds, if the Base ABCD (Fig. 38.) being divided into little square Area's, be multiplied by the heigth AE, divided by a like measure for length, after this way you may conceive as many equal little Cubes to be generated in the whole Solid, as is the number of the little Area's of the Base multiplied by the number of Divi∣sions of the side AE; you may moreover obtain the Solidity of any other Parallelepipeds, that are not Right Angled ones, by multiplying their Bases and Perpendicular Heigths together.

IV. Moreover since every Triangular Prism is the half of a Paral∣lelepiped, and any Multangular Prism may be resolved into as ma∣ny Triangular ones, as its Base contains Triangles; you may obtain the Solidity (or Solid Contents) either of the one or the other, if you multiply the Triangular, or Multangular Base of them into their Perpendicular Heigth.

V. After the same manner you may likewise have the Soli∣dity of a Cylinder, which may be considered as an Infinite Angled Prism, just as the Circle is as an Infinit-Angled Po∣lygon.

DEFINITION XVII.

IF any Triangle ABC (Fig. 39. N. 1.) be conceived to move with one of its Plane Angles C, from the Vertex or top of a Solid Angle (determined by two Planes aAb and cAa joined together in the common Line Aa) with a motion always parallel to it self; so that its extreme Angular Point A shall al∣ways remain in the Line Aa, but with its Sides AB and AC shall all along raze on the two Angular Planes, till at length it falls wholly within the Solid Angle: by this its motion it will describe within the Solid Angle, the Figure we call Pyramidal, whose Base will be the Triangle abc, and its Vertex A will also describe without it another Quadrangular Pyramid, whose common Vertex will be the same A, but the Base the Quadrangle Cb, described by the Side of the moveable Triangle BC: The first Py∣ramid it will describe with its Triangular Parts, 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉 continually in∣creasing

from the point A, and ending in the Triangle abc; but the latter Pyramid will be described by the remaining parts, continually decreasing downwards from the whole ABC, and the Quadrangular Trapeiza 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉 at length ending in the Right Line bc: So that in the mean while the said Triangle with its whole space describes, according to what we have said before, the Triangular Prism composed of those two Pyramids. From this Geeesis of Pyramids you'l have the following

CONSECTARYS.

I. OF what sort soever the Describing Triangles are ABC, ABC (N. 2.) fo they are equal; and whatever the Solid Angles are, comprehended under the Planes abA, and acA, abA and acA, so they are accommodated to the Plane Angles A and A, and such that, &c.

DEFINITION XVIII.

THere may be exhibited another easier Genesis of Cones and Pyramids, but it respects only the Dimension of the Surface, and not of the Solidity of them, viz. If you have a fix'd point A that is not in the Angular Plane BCDEF (Fig. 41.) and a Rght Line AF let fall from that point to any Angle of the Plane, be con∣ceived to move round the sides BC, CD, &c. This Plane by its motion will describe as many Triangles ABC, CAD, &c. as the Angular Plane has Sides. And these Triangles all meet∣ing at the point A, make that Solid which we call a Pyramid. Now if instead of the Angular Plane there be supposed a Cir∣cular one, (or an Angular one of Infinite Sides) the Solid thence produc'd is called a Cone, whose Surface is equal to Infinite Tri∣angles, constituted on the Base BCDE, and whose Solidity would consequently equal an Infinite Angled Pyramid of the same heigth. And after the same manner by the motion of the Line AF, remaining always Parallel to it self about Parallelo∣grams or Triangular Planes, will be generated Parallelipipeds, Prisms, and Cylinders. But as one Pyramid will be produced more upright than another, according as the point A stands more over the middle of the Plane BCD, &c. (Fig. 42.) or

[illustration]

Pag. 28.

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Schemes for rai-sing the five Regular Bodys

respects it more obliquely: So in particular a Cone is called a right Cone, when the line AO being let fall to the Center of the circular Plane (otherwise call'd the Axis of the Cone) constitutes on all sides right Angles with it; but it is call'd an Oblique or Sca∣lene Cone, when the Ax stands obliquely on the Base. Which distinction may also be easily understood when apply'd to Cylin∣ders, tho' a right Cone and Cylinder may also be conceiv'd to be generated after another way, as if for the one a Triangle and for the other a right angled Parallelogram AOB and LAOB be be conceiv'd to be moved round a line AO considered as immoveable (whence it is call'd an Ax;) and also a truncated Cone may be formed if a right angled Trapezium,(a) 1.40 2 of whose Sides are parallel be moved, &c. And as we have deduc'd the So∣lidity of these Bodies from the foregoing Genesis, so their external Surfaces, as also of Prisms and Parallelepipeds may easily be found from the present Genesis, by any one who attentively considers the following

COROLLARYS.

I. SInce the whole external Surface, except the Base, of any Pyramid is nothing but a System of as many Triangles ABC, CAD &c. as the Pyramid has Sides; if the Area's of those Triangles separately found by Consect. 8. Def. 12. be added into one Sum, you'l have the Superficial Area of the whole Pyramid.

II. If a Pyramid be cut with a Plane b, c, d, e, parallel to its Base BCDE (Fig. 41.) The Surface of that truncated Py∣ramid comprehended between the prallel Planes may be obtain'd if having found the Surface of the Pyramid A bcde cut off from the rest by Consect. 1. you subtract it from the Surface of the whole Pyramid.

II. The external Surface of a right Pyramid that stands on a regular Polygon Base is equal to a Triangle, whose Altitude is equal to the Altitude of one of the Triangles which compose it, and its Base to the whole Circumference of the Base of the Pyramid.

IV. Therefore the Surface of a right Cone, by what we have already said, is equal to a Triangle whose heighth is the

side of the Cone, and the Base equal to the Circumference of the Base of the Cone.

V. The Surface of a truncated right Cone, or Pyramid is equal to a Trapezium which has 2 parallel Sides the lowest of which is equal to the Perifery of the Base, and the other to the Perifery of the Top or upper Part, and the heigth, to the inter∣cepted Part.

VI. The Surface of a right Cylinder or Prism is equal to a Parallelogram which has the same height with them, and for its Base a right line equal to the Perifery of that Cylinder or Prism.

DEFINITION XIX.

IF a semicircular Plane ADB (Fig. 43. N. 1.) be conceived to move round its Diameter AB which is fixt, as an Axis, by this Motion it will describe a Sphere, and with its Semicircum∣ference the Surface of that Sphere; every part whereof is equally distant from the middle Point of that Axis C (which is there∣fore call'd the Center of that Sphere.) Now if (N. 2.) this semicircular Ambitus(a) 1.41 be conceived to be divi∣ded before that revolution, first into 2 Quadrant•• AD, and BD, and then each of those again into as many parts equal in Number and Magnitude as you please, and having drawn the Chords AF, FE, ED, &c. let the Polygon AFEDGHB In∣scribed in the Semicircle be conceived together with it to be turn'd about the Axis AB; then will A 1 F, and B 4 F de∣scribe 2 Cones about the Diameters F f, and H h; and the Tra∣pezia about the Axes 1, 2, 2 C, C 3, and 34, will describe so many truncated Cones, and the lines AF, FE, &c. so many Conical Superficies, by the Antecedent Def. and so the whole Po∣lygonal Plane AFEDGHB a Conical Body inscribed in the Sphere, and contain'd under only Conical Surfaces. And as any attentive Person may easily perceive such a Body to be less than the ambient Sphere, and its whole Surface less than the Surface o•• the ambient Sphere; so he may as easily trace these following

CONSECTAYS.

I. IF the Arches AF, FE, &c. be further bisected, and a Polygonal Figure of double the number of Sides inscri∣bed in the Semicircle, and conceived to be moved round after the way we have already shewn, the Pseudoconical Body hence arising, will approach nearer and nearer to the Solidity of the Sphere, and the Surface of the one to the Surface of the other, and hence (if we continue this Bisection, or conceive it to be continued ad Infinitum) you may infer.

II. That a Sphere may be look'd upon as much a Pseudoco∣nical Body, consisting of infinite Sides, and it's Surface will be equal to the infinite Conical Surfaces of that Body; which we will take further notice of below.

DEFINITION XX.

IF the Diameter AB of the Semicircular Plane ADB (Fig. 43. N. 3.) be conceived to be divided into equal Parts (as here the Semidiameter AC into 3.) and if the circumscribing Pa∣rallelograms CE, 2 E, 1 G on the transverse Parallels CD, 2 e, 1 f be conceived together with the Semicircle it self to revolve about the fixed Ax AB; it is evident that there will be formed from the Semicircle a Sphere as before, and from the Circum∣scribed Parallelograms, so many circumscribed Cylinders of e∣qual heighth: but if all the Altitudes or Heighths of these are bisected or divided into two, and so make the number of circum∣scribing Parallelograms double, there will be formed (by mo∣ving them round as before) double the Number of Cylinders of half the heighth, but which yet being taken together, approach much nearer the solidity and roundness of the Sphere, than the for∣mer, which were fewer in Number (viz. the six latter Parallelo∣grams approach nearer to the Plane of the Circle than the three former) and thus if that bisection of the Altitudes be conceived to be continued ad infinitum, the innumerable Number of those in∣finitely little Cylinders will coincide with the Sphere it self. More∣over if you conceive any Polyedrous or Multilateral Figure to be circumscribed about the Sphere (which we here endeavour to de∣lineate by the Polygon ABCD N.4. circumscribed about the Cir∣cle)

and the solid Angles thereof to be cut by other Planes ab, which shall touch the Sphere; it is manifest there will thence a∣rise another Polyedrous Figure, the Solidity whereof will approach nearer to the Solidity of the Sphere, and its Surface to the Sphe∣rical Surface than the former, and if the Angles of this be a∣gain in like manner cut off, there will still arise another new So∣lid, and new Surface approaching yet nearer to the Solidity and Surface of the Sphere than the former, &c. and so after an infi∣nite Process they will coincide with the Sphere and its Surface themselves. Whence flow these

COROLLARYS.

I. THE Sphere may be considered as a Polyedrous Figure, or as consisting of innumerable Bases, i. e. composed of an innumerable Number of Pyramids, all whose Vertex's meet in the Center, and so whose common heigth is the Semidia∣meter of the Sphere, and the sum of all the Bases equal to the Superficies of the Sphere.

II. If you can find a Proportion between a Cylinder of the same heigth with any Sphere, and whose Base is equal to the greatest Circle of that Sphere, and innumerable Cylinders cir∣cumscribed about it, as we have just now shewn; then you may also obtain the Proportion between the said circumscrib'd Cylinder and the inscrib'd Sphere: Which to have here hinted may be of service hereafter in its proper place.

DEFINITION XXI.

THere remain those Bodies to be consider'd which are call'd Regular, which correspond to the Regular Plane Figures; and as those consist of equal Lines and Angles, so these likewise are comprehended under Regular and Equal Planes meeting in equal solid Angles; and as those may be Inscribed and Circum∣scribed about a Circle, so may the latter likewise in and about a Sphere. But whereas there are infinite Species of Regular Plane Figures, there are only five of Regular Solids; the first whereof is contained under four Equal and Equilateral Trian∣gles, whence it is nam'd a Tetraedrum; the second is terminated by six equal Squares, and thence is call'd Hexaedrum, and other∣wise

a Cube; the third being comprehended under eight Equal and Equilateral Triangles, is call'd an Octaëdrum; the fourth is contained under twelve Regular and Equal Pentagons, and so is nam'd a Dodecaëdrum; the fifth, lastly, is contained under twenty Regular and Equal Triangles, and is thence nominated an Icosa∣ëdrum. Besides these five sorts of Regular Bodies there can be no other; for from the concourse of three Equilateral Triangles arises the Solid Angle of a Tetraëdrum, from four the Solid An∣gle of an Octaëdrum, from five the Solid Angle of an Icosaëdrum; from the concourse of four Squares you have the Solid Angle of an Hexaëdrum; from that of three Pentagons you have the Solid Angle of a Dodecaëdrum; and in all this Collection of Plane Angles, the Sum does not arise so high as to four Right ones. But four Squares, or three Hexagons meeting in one Point, make precisely four Right Angles, and so by Consect. 2. Definit 8. would constitute a Plane Surface, and not a Solid Angle. Much less therefore could three Heptagons or Octagons, or four Pentagons meet in a Solid Angle, to form a new Regular Body; for those added together would be greater than four Right An∣gles. But now, for the Measures of these five Regular Bodies, take the three following

CONSECTARYS.

I. SInce a Tetraëdrum is nothing else but a Triangular Py∣ramid, and an Octaëdrum a double Quadrangular one, their Dimension is the same as of the Pyramids in Schol. of De∣finit. 17.

II. The Solidity of an Hexaëdrum may be had from Consect. 3. Definit. 13.

III. A Dodecaëdrum consists of twelve Quinquangular Py∣ramids, and an Icosaëdrum of twenty Triangular ones, all the Vertex's or tops whereof meet in the Center of a Sphere that is conceived to circumscribe the respective Solids, and consequent∣ly they have their Altitudes and Bases equal: Wherefore ha∣ving found the Solidity of one of those Pyramids, and multi∣plied it by the number of Bases (in the one Solid 12, in the o∣ther 20) you have the Solidity of the whole respective Solids.

DEFINITION XXII.

BEsides these Definitions of the Regular Bodies, we may al∣so form like Idea's of them from their Genesis, which particularly Honoratus Fabri has given us a short and ingenious System of, in his Synopsis Geometrica, p. 149. and the follow∣ing.

I. Suppose an Equilateral Triangle ABD to be inscrib'd in a Circle (Fig. 44. N. 1.) whose Center is C, whence having con∣ceived the Radii CA, CB, CD, to be drawn, imagine them to be lifted up together with the common Center C, so that the point C ascending Perpendicularly, at length you'l have the Line•• EA, EB, ED, equal to the Lines AB, BD, DA, After this way there will be generated, or made a Space consisting of fou•• Equal and Equilateral Triangles, which is call'd a Tetraëdrum Hence we shall by and by easily demonstrate, the quantity o•• the Elevation CE, and the Proportion of the Diameter of th•• Sphere EF to be Circumscribed to the remaining part CF and so the reason of the Euclidean Genesis proposed lib. 13 Prop. 13.

II. Much like this, but somewat easier to be conceived, is th•• Genesis of the Octaëdrum, where by a mental raising of the Cen∣ter C (Fig. 44. N. 2.) of the Square ABDE inscribed in th•• Circle, together with the Semidiameters CA, CB, CD, CE until being more and more extended they at length become th•• Lines AF, BF, DF, EF, all equal among themselves, and 〈◊〉〈◊〉 the side of the Square AB or BD; and its manifest, that by th•• like extension conceived to be made downwards to G, the•• will be formed eight equal and regular Triangles, which w•• all concur in the two opposite Points F and G. We migh•• also deduce another Genesis of the Octaëdrum from a certai•• Section of a Sphere, and also give the like of a Hexaëdrum o•• Cube: but we have already given the easiest, of the one, vi•• that which is also common to Parallelepipeds; and that of th•• other just now given is sufficient to our purpose.

Notes

• (a) 1.1

  Eucl. Ax. 14.
• (a) 1.2

  Eucl. Ax. 14.
• (a) 1.3

  Eucl. l. 1. Prop. 20.
• (b) 1.4

  Eucl. Prop. 2. lib. 3.
• (b) 1.5

  & 6 of the same Book.
• (c) 1.6

  Eucl. 8. l. 3.
• (a) 1.7

  Eucl. 5 l. 3
• (a) 1.8

  Prop. 15. lib. 3.
• (a) 1.9

  Eucl. 13. lib. I. with the Coroll.
• (a) 1.10

  Euclid. l. 1. Prop. 14.
• (b) 1.11

  Eucl. l. 1. 4 & 3.
• (c) 1.12

  lib. III. 3.
• (d) 1.13

  lib. 1. 5.
• (a) 1.14

  Eucl. lib. 1. 15.
• (a) 1.15

  The same Prop. Schol. 1.
• (b) 1.16

  Schol. 2.
• (a) 1.17

  Among the rest Vid. Eucl. 28 & 29 lib. 3, and also part∣ly the 26 and 27.
• (a) 1.18

  Eucl. p. 20. l. 3.
• (b) 1.19

  Eucl. p. 9. l. 1.
• (a) 1.20

  Eucl. 31. lib. 2.
• (b) 1.21

  Eucl. 26 and 27. lib. 3.
• (a) 1.22

  Eucl. lib. 1. 30.
• (b) 1.23

  Eucl. lib. 1. 29.
• (a) 1.24

  lib. 1. Prop. 27 & 28
• (a) 1.25

  lib. 1. Prop. 32,
• (a) 1.26

  Schol. Prop. 34. lib. I.
• (b) 1.27

  The first part of the same Proposit.
• (c) 1.28

  the latter part of the same Prop.
• (a) 1.29

  lib. 1. Prop. 35 & 36
• (a) 1.30

  lib. 1. Prop. 37 & 38, & 41.
• (b) 1.31

  lib. I. Prop. 32. otherwise.
• * 1.32

  This is ma∣nifest from its Genesis; for the five parts of the Line AB by its motion along the part of the Base BE, de∣scribes 5 little Squares, and moving along the following part, describes five more, &c.
• (a) 1.33

  lib. I. Prop. 5. the same otherwise demonstrated.
• (a) 1.34

  Archimedes of the Dimen∣sion of the Circle, Prop. 1.
• (a) 1.35

  Euclid. Prop. 15. lib. 4.
• (b) 1.36

  Eucl. 2d Coroll. of the same.
• (c) 1.37

  Eucl. 6. lib. 4.
• (a) 1.38

  Eucl. l. 11. p. 29, 30, 31
• (b) 1.39

  Eucl. l. 11. p. 28.
• (a) 1.40

  Archimedes lib. I de Con. and Cylin. prop. 7. and 8.
• (a) 1.41

  Archimedes lib. I. de Cono & Cyl. Prop. 22. Coroll. & Prop. 23.