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.

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.