Let the former series be drawn into another series of x, and there will be produced x(d + x)x2 = x(dx2 + x3) + xx (dx2+x3)4=x(2x2+1+x)=(2dx2+1+x), hence, we have

=

(2dr* + 1+x) 4dx + 3x2 (3dx2 + 2x' 6d+ 4x the vertex.

the distance of the centre from

Problem. 11. To find the virtual centre of the arc of a circle.

The virtual centre of an arc of a circle, is the same in reference to the centre of the circle, as that of the segment of a revoloidal curve, whose conjugate diameter is the same as that of the circle, and the base of whose segment is equal to the given arc of the circle.

For, if a revoloidal

E

nate fg will be equal to the arc FEG. Draw across the segment fEgf, equidistant ordinates, perpendicular to fg, and they will represent the distance of the several points in the arc, from the line fg, since they are supposed to be drawn from points equidistant from each other on the line fg, or the arc FEG; and, since if there is an infinite number of ordinates, they may be regarded as the area of the segment, it therefore, follows that if this area is divided by the arc, the quotient is the distance of the centre, from the line FG; also, if the area fEgHD is divided by the arc FEG or line fg, the quotient is the virtual centre of the arc fEg from the axis DH.

Let c = the chord FGa = the arc FEG, and r = CE, then will cr = area DfEgH, (Prop. III, Cor. 4, B. III.,) hence, we

I cr have = the distance EI of the centre.

a

This is also the virtual centre of the segment f Egf of the revoloidal curve, and it may also be shown that if a series of equidistant ordinates to the axis CE, are drawn through the revoidal surface, or any segment of it parallel to fg, the series of ordinates so constructed, drawn into their several distances from any given line, parallel to such ordinates, will determine the virtual centre of the segment fgef, or of the arc FEG, by proceeding as before.

Art. 12. If it be required to find the virtual centre of the arc of an ellipse, a parabola, or an hyperbola, it may be done in a similar manner, by taking a portion of the surface of the elliptic, parabolic, or hyperbolic revoloid, and proceeding as for the arc of the circle, which also gives the virtual centre of the segment of the revoloidal surface pertaining thereto.

Problem 3. To find the virtual centre of the surface of a solid.

Let the proposed surface be the convex surface of a pyramid ABCDV; and because any portion of the convex surface, included between any two sections, by planes parallel to the base, is proportional to the portion of a vertical triangle through the pyramid included between the same planes; it follows that the vertual centre of the convex surface, is the same as that of the vertual triangle; hence the same process will determine both. If it is required to find the virtual centre of the whole surface of a pyramid, including its base, we have only to imagine an infinite number of ordinates to be drawn across the several triangular sides parallel to their several bases, and also a similar series of parallel ordinates across the base, and if each of these ordinates are severally drawn into their respective perpendicular distances from the vertex of the given pyramid, we shall have produced, as many

new pyramids AEFCQ, whose bases ACFE, ABHG, &c, are, severally equal to the bases of the sides of the pyramid multiplied by IA, the distance of the base from the vertex, as the given pyramid has sides, and also a prism ABDCQIKR, formed by drawing every line in the base, or the whole surface of the base into the distance of the base from the vertex.

Q

R

K

PN

F

E

B

G

H

And the sum of the imaginary solids so generated, divided by the whole surface of the pyramid, will give the distance of the virtual centre from the vertex.

Let AB=x and if the pyramid is generated from a or hx, or if the base of the pyramid is a square, the perimeter of the base will be 4x ; and since each pyramid ACFEQ, is equal to

3

the prism ABCDQRKI: hence the four pyramids generated by the series drawn into the four sides of the base are = the prism, and if h = the altitude IA of the pyramid, hx2+hr2= thx = the sum of the four pyramids + the prism ABDCQIKR;

the surface of the given pyramid is = x2+2x√(h2+x2)

Hence we have,

Thx

Thx

x+2√h2+x2

equal the distance of the virtual centre from the vertex.

BOOK VII.

CHAPTER V.

ON THE RELATIONS OF LINES, SURFACES, AND SOLIDS,
GENERATED BY MOTION.

THE capacity of any solid generated by the motion of a surface perpendicular to itself, is measured by the generating surface drawn into the distance moved; which distance is always equal to the distance passed through by the virtual centre of such surface.

If the motion of the generating surface is such, as that it always maintains a parallel position, and moves in a direction perpendicular to itself, the proposition is sufficiently manifest.

Let now the rectangle ACBD revolve about the side BC, which remains fixed, and the product will be the cylinder DF, whose solidity is equal to the surface ACBD drawn into the circumference PK, described by the virtual centre K, of the plane, which centre is in this case also the centre of aggregation.

DdB

A

D

K

a

B

L

P

N

G

F

M

If a right-angled triangle ABC revolve about the perpendicular BC, so as to describe the cone ABD, this is also measured by the triangle ABC drawn into the circumference FL, described by the virtual centre of the triangle. The virtual centre of the triangle we have shown to be situated at the point F, on the line BE, from the vertex bisecting the base at a distance from B = its length. Let A the surface ABC be multiplied by the circumference described by the centre F; and since the radius FG=EC, hence the circumference FG= the circumference EC, and because the triangle ABC=1 its circumscribing rectangle ADBC, which generates a cylinder ADNM, the generating surface of the triangle ABC drawn into the circumference, is equal to one-third the cylinder generated by the rectangle, or one-third the rectangle drawn into the circumference EC, as it ought to be.

And in general, let any plane figure be revolved about any line or axis without the figure, but always in the same plane, and the solid generated will be measured by the generating surface drawn into the arc described by the virtual centre of the surface.

S

P

M

L

R

Let AFHD be a solid generated by the plane ABD; through C, the virtual centre of which, draw DCAE, perpendicular to the axis of rotation, and meeting HGFE in E, let an indefinite number of parallel ordinates, ef, ik, &c., be drawn across the generating surface, parallel to the axis about which it revolves; and the solid generated is equal to all of those ordinates, drawn into the distances passed through by each; viz., the ordinate drawn across the point Ax the arc AF+the ordinate ab drawn through cx the arc CG+, &c., through the whole series. And because EA, EC, ED, &c., are as the arcs AF, CG, DH, &c. Hence Ehxef, and Elxik, &c., are as phxef, and rlik, and because EC drawn into all the ef, ik, &c., is equal to all, the Ehxef, Elxik, &c., it follows that CGxall the ef, ik, &c., is equal to all the ph×ef, rlik, &c.; or that the solid ABDHF is equal to the generating surface ABDe drawn into the line described by the virtual centre of the surface.

E

A MEND

f B

2

Cor. 1. Hence, if any curve or any line be made to revolve about any axis exterior to such curve, but in the same plane, the surface described by its motion will be equal to the line or curve drawn into the distance passed through by the virtual centre of such line or curve.

For, let the perimeter of the figure generating the solid above, be the generating line, and let us suppose its virtual centre the same as before; let every point in this perimeter be reduced to the line AD by means of perpendiculars thereto; and the figure generated by its revolution about the axis, is equal to all the ph, rl, &c., described by every point; but we have seen that all the ph, rl, &c., are as all the Eh, El, &c.; and since the sum of all the Eh, El, &c., is equal to as many times EC, therefore the sum of all the ph, rl, &c., is equal to as many times CG, or equal to ABDeXCG, that is, the surface described by the perimeter ABDe, is equal to ABDe drawn into the line described by its virtual centre C.

Cor. 2. From E draw EIKL, cutting the upright prismatic figure erected on the given base ABD, so as that any perpen

dicular AI may be equal to the corresponding arc AF. Then will the figure AILD be equal to the figure AFHD.

For, by similar figures, all the AF, CG, DH, &c., are as all the AI, CK, DL, &c., each to each; and as one of each are equal, therefore they are all equal, each to each; viz., all the AI, CK, DL, &c., equal to all the AF, CG, DH, &c.; that is, the figure AILD equal to the figure AFHD.

Cor. 3. Through K draw MKNO; then the figure ANMD will be equal to the figure AIKLD, or equal to the figure AFHD.

For, by the last corallary, AFMD is equal to the figure described by the base AD, revolving about O, till the arc described by C be equal to CK; which, by the proposition, is equal to ADXCK, or ADXCG.

Cor. 4. Hence, all the upright figures AQKRD, AIKL, ANKMD, AKPD, &c., of the same base, and bounded at the top by lines or planes cutting the upright sides, and passing through the extremity K, of the line CK, erected on the virtual centre of the base, are equal to one another; and the value of each will be equal to the base drawn into the line CK.

Hence, also, all figures described by the rotation of the same line or plane about different centres or axes, will be equal to one another, when the arcs described by the virtual centre are equal. But if those arcs be not equal, the figures generated will be as the arcs. And in general, the figures generated, will be to one another, as the revolving lines or planes drawn into the arcs described by their respective virtual

centres.

Cor. 5. Moreover, the opposite parts NIK, MLK, of any two of these figures, are equal to each other.

Cor. 6. The figure ASPD is to the figure APD, as AS to CK; for, by similar triangles, they will be as AD to AC.

For ASPD is equal to ADXAS, and APD equal to AD XCK.

Cor. 7. If the line or plane be supposed to be at an infinite distance from the centre about which it revolves, the figure generated will be an upright surface or prism, the altitude being the line described by the virtual centre; so that the base drawn into the said line will be equal to the base drawn into the altitude, as it ought for all upright figures, whose sections parallel to the base are all equal to each other.