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EA, EC, ED, &c; and, by mechanics, the fum of all the EA, EC, ED, &c, is equal to as many times EC; therefore the fum of all the AF, CG, DH, &c, is equal to as many times CG, or equal to AD X CG; that is, the figure AFHD is equal to ABD X CG, the base drawn into the line defcribed by its center of gravity. 2. E.D.

COROLLARY 1.

From E draw EIKL Cutting the upright prifmatic figure erected upon the given bafe ABD, fo as that any perpendicular AI may be equal to its correfponding arc AF. Then will the figure AILD be equal to the figure AFHD.

For, by fimilar 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, there

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fore 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.

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Through к draw мKNO; and the figure ANMD will be equal to the figure AIKLD, or equal to the figure AFHD.

For, by the laft corollary, ANMD is equal to the figure defcribed by the bafe AD revolving about o, till the arc defcribed by c be equal to ck; which, by the propofition, is equal to AD X CK OF AD X CG.

COROL

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Hence all the upright figures AQKRD, AIKLD, ANKMD, AKPD, &c, of the fame base, and bounded at the top by lines or planes cutting the upright fides,' and paffing through the extremity K of the line cK erected upon the center of gravity of the bafe, are equal to one another; and the value of each will be equal to the bafe drawn into the line cк.

Hence alfo all figures, defcribed by the revolution of the fame line or plane about different centers or axes, will be equal to one another, when the arcs defcribed by the center of gravity 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 defcribed by their refpective centers of gravity.

COROLLARY IV.

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

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The figure ASPD is to the figure APD, as As to CK; or, by fimilar triangles, they will be as AD

to Ac.

For ASPD is equal to AD X AS, and APD is equal

to AD X CK.

COROLLARY

VI.

If the line or plane be fuppofed to be at an infinite distance from the center about which it revolves, the figure generated will be an upright furface or prifm, the altitude being the line defcribed by the center of gravity; fo that the bafe drawn into the faid line will

K k 4

be

be equal to the base drawn into the altitude, as it ought for all upright figures, whose sections parallel to the bafe are all equal to each other.

EXAMPLE I.

If a right line, or a parallelogram, revolve about a line perpendicular to the length, there will be defcribed a ring either fuperficial or folid; and as the center of gravity of the describing line, or parallelogram, is in the middle of them, the general rule will become the fame with rule 3 fect. 1 part 3, and the rule at prob. 2 fect. 3 part 3.

When the center of revolution is in the end of the line, the line will defcribe a circle whofe radius is the faid defcribing line, and whofe circumference is double the circumference defcribed by the center of gravity; confequently the radius drawn into half the circumference, will be the area of the circle.

EXAMPLE II.

If the right-angled triangle ABC

revolve about the perpendicular BC, and defcribe the cone ABD.

Draw BE to bifect Ac, and take

EF equal to one-third of BE; and

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F will be the center of gravity of the triangle ABC, as is well known. Draw FG parallel to AC.-Then the line defcribed by F will be the circumference of a circle whofe radius is FG; and, by the general rule, the cone will be equal to the triangle ABC X GF X 8n, 'putting n = 785398 &c, or AC XCBX÷CE X 8n AC2 X÷CB × 4n = AD3

AC XCBX÷CA X En = XCB x n = the base drawn into one-third of the altitude, as it ought.

Again, from H, the middle of AB, draw HI parallel to AC; then is H the center of gravity of AB, and confequently the furface defcribed by AB will be AB X circumference whofe radius is HIAB X half the circumference whofe radius is AC the fide drawn into half the circumference of the base the furface of the cone, as it ought.

EXAMPLE III.

Let the femi-circle DCA revolve about the diameter AD, and describe the furface of a sphere.

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If there be taken DC: FC :: FC: FH ==

FC2
DC

= 477

putting r for the radius, and c for the whole circumference; H will be the center of gravity of the arc DCA, and confequently rc:: FH: 4r the line or circumference described by н the center of gravity; and, by the general rule, DCA X 4r 1 cx 4r = 2rc = the furface of the fphere the circumference into the diameter, as it ought.

And for the folidity of the sphere, we fhall have -the distance FH of the

8rr
3C

first 3c: 2r :: 3r: 30

center of gravity of the femi-circle DCAD from the diameter AD, which is two-thirds of the distance of

the

the center of gravity of the arc DCA from the fame diameter DA, in the former cafe; confequently the line described by the center of gravity in this cafe will be two-thirds of that in the former; but the defcribing line in the former cafe is to the defcribing fpace in this, as 1 is to r, therefore 1: × r:: furface of the fphere: folidity = r × surface.

COROLLARY.

The circumference of the circle, whofe radius is the distance of the center of gravity of the femi-circumference of any circle from its center, is equal to four times the radius of that circle.

EXAMPLE IV.

For the folidity of the parabolic fpindle, putting b the bafe and a the altitude or axe of the rating parabola, and n = 785398, as before.

I 6

gene

It is known that 3a is the distance of the center of gravity from the bafe, and confequently a = the line defcribed by the center of gravity; but ab is the revolving area; therefore a xab: aab will be the content, which is of the circumfcribed cylinder.

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For the paraboloid. Making the notation as in the laft example, 3b will be the distance of the center of gravity of the femi-parabola from the axe, confequently 36 Sn × 3ab = 2abbn the folidity -half the circumfcribed cylinder.

=

PART

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