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Whatever presses or draws another, is as much pressed or drawn by that other. If any one, with his finger press a stone, his finger is also pressed by the stone. If a horse draw a stone tied to a rope, the horse, if I may so speak, will be equally drawn back towards the stone : for the stretched rope, by the endeavour of relaxing itself, will urge the horse towards the stone, and the stone towards the horse, and will as much impede the progress of one, as it advances that of the other. globular body, as an ivory ball, impinging on another similar one, by its force change in any way the motion of that other, the same will also by the force of that other, on account of the equality of the mutual pressure, undergo an equal change in its motion, to the contrary part. By these actions, if the bodies be unequal, equal changes are made, not of velocities, but of motions ; namely, in bodies not otherwise obstructed: for the changes of velocities, made towards contrary parts, because the motions are equally changed, are reciprocally proportional to the magnitudes of the bodies.

In attractions, which are the principal object of this

A

B В tract, the truth of this law may be thus shewn. Between any

D

G two bodies A and B, mutually attracting each other, conceive any obstacle to be placed, by

E

II which their coming together may be hindered. If either body A be more attracted towards the other B, than that other B towards the former A, the ob

K stacle would be more urged by the pressure of the body A, than by that of B, and therefore would not remain in an equilibrium. The stronger pressure would prevail, and cause, that the system of the two bodies and the obstacle would be moved directly towards the part, on which В lies, and in free spaces would go forward in infinitum, with a motion continually accelerated; which is absurd and contrary to the first law. For, by the first law, the system ought to perseverè in its state of rest, or of moving uniformly forward in a right line; therefore the bodies must equally press the obstacle, and are equally attracted by each other. The truth of this may be shewn by experiment, in the attraction between a magnet and iron. If these, placed apart in proper vessels touching each other, float near each other in still water, 'neither will propel the other, but, by the equality of attraction both ways, they will sustain each other's pressure, and at length rest in an equilibrium.

So also the gravity between the earth and its parts is mutual. Let the earth EH be cut by any plain DF into two unequal parts DEF and DHF; their weights towards each other are mutually equal. For if the greater part DHF be, by another plane GK parallel to the former DF, cut into two parts DFKG and GHK, of which the exterior part GHK is equal to the less part first cut off DEF: it is manifest, that the middle part DFKG will, by its own weight, tend to neither of the extreme parts, but will, if I may so speak, be suspended, and rest in an equilibrium between both. But the extreme part GHK would press with all its weight on the middle part, and urge it towards the other extreme part DEF; therefore the force with which the sum of the parts GHK and DFKG tends towards the third part DEF is equal to the weight of the part GHK, or, to the weight of the third part DEF. Therefore the weights of the two parts DEF and DHF towards each other are equal, as was proposed to be proved. And unless these weights were equal, the whole earth, floating in a free ether, would yield to the greater weight, and, in going from it, would be carried off in infinitum.

A body, by two conjoined forces, describes the diagonal of a parallelogram, in the same time, in which it would describe the sides, by them separately.

If a body, in a given time, by the force Malone, im

B pressed on it in the place A, be borne with a uniform motion from A to B ; and by the force N alone, impressed

D on it in the same place, be borne, in the same time, from A to C; the parallelogram ABDC being completed, and the diagonal AD drawn, the body by both forces acting together, would, in the same time, be borne, with a uniform motion, in the diagonal AD.

For because the force N acts in a direction AC parallel to BD, this force, by law 2, will nothing alter the velocity of approaching to the right line BD, generated by the other force ; the body will therefore arrive at the right line BD, in the same time, whether the force N be impressed on it, or not; and therefore, at the end of that time, will be found somewhere in the right line BD. By a similar argument, it will, at the end of the same time, be found somewhere in the right line CD, and therefore in the concourse D of BD and CD. And since, if through any point whatever in AD, right lines be drawn to AC and AB, parallel to AB and AC, proportional parts would be cut off from AB, AC and AD ; it may by a like argument be proved, that in any part of the given time, the body would describe a part of AD, having the same ratio to AD, as the part of the time to the whole, therefore the body is borne with a uniform motion in AD.

Scholium. From this corollary follows, the composition of a direct force AD, from two oblique ones AB and BD ; and on the contrary, the resolution of any direct force AD, into two oblique ones AB and BD; which composition and resolution s abundantly confirmed from mechanicks.

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LEMMA I.- See Note.

Quantities and the ratios of quantities, which tend continually to

equality, so as at length to differ from each other less, than by any given difference, are ultimately equal.

If not, let them be ultimately unequal, and let their ultimate difference be D. Therefore they cannot approach nearer to equality, than by the given difference D; contrary to the supposition.

LEMMA II.

If in any figure JacE bounded by two right lines Aa and AE at

right angles to each other, and a curve line acE, any number of rectangles Ab, Bc, Cd, foc. be inscribed, contained under equal bases AB, BC, CD, doc. and sides Bb, Cc, Dd, &c. parallel to the side ja of the figure, and the parallelograms a Kbl, bLem, Mon, f'c. be completed, and if the breadth of these parallelograms be diminished, and their number increased in infinitum ; the ultimate ratios, which the inscribed figure AKILMI, the circumscribed figure Aalbmondo E, and the curvilincal figure Aabcd E, have to each other, are ratios of equality.

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For the difference of the inscribed and circumscribed figures is the sum of the parallelograms K), Lm, Mn, Do, or, which is, because of the equal bases of all, equal, the rectangle under the base Kb of one, and the sum of the altitudes Aa, or the rectangle A Bla; but this rectangle, because its breadth AB is supposed to be diminished in infinitum, becomes less than any given space; therefore, (by Lemma 1,) the inscribed and circumscribed figures, and much more the intermediate curvilineal figure, become ultimately equal.

A B C D

LEMMA III.

The same ultimate ratios, are also ratios of equality, when the

breadths AB, BC, CD, d'c. are unequal, and are all diminished in infinitum.

For let AF be equal to the greatest breadth, and let the parallelogram AFfa be completed ; this is greater than the difference of the inscribed and circumscribed figures ; but its breadth being diminished in infinitum, it becomes at length less than any given rectangle.

Cor. 1. Hence the ultimate sum of these evanescent parallelograms, coincides in every part with the curvilineal figure.

Cor. 2. And much more, the rectilineal figure, which is comprehended under the chords of the evanescent arches ab, bc, cd, &c. coincides ultimately with the curvilineal figure.

Cor. 3. As also the circumscribed rectilineal figure, comprehended under the tangents of the same arches.

Cor. 4. And therefore these ultimate figures, (as to their perimeters acE, are not rectilineal, but curvilineal limits of rectilineal figures.

LEMMA IV.

If in two figures there be inscribed, as in the preceding lemma, two

ranks of parallelograms, an equal number in each figure, and, when their breadths are diminished in infinitum, the ultimate ratios of the parallelograms in one figure to those in the other, each to each, be the same ; these tuo figures are to each other, in the same ratio

For as the parallelograms in one figure are to those in the other, each to each, so is the sum of all the parallelograms in the former to the sum of all in the other (12. 5 Eu.), and so is the former figure to the other, the former figure being to the former sum, and the latter figure to the latter sum, in the ratio of equality (by Lemma 3).

Cor. Hence if two quantities of any kind, be any how divided into an equal number of parts, and these parts, when their number is increased, and magnitude diminished in infinitum, have a given ratio to each other, the first to the first, the second to the · second, and the others in their order to the others; the whole quantities are to each other in the same given ratio. For if, in two such figures, as those mentioned in this lemma, parallelograms be taken, which are to each other as the parts, the sum of the parts are always as the sum of the parallelograms (12 and 11. 5 Eu.), and therefore, when the number of the parts and parallelograms is increased and their magnitude diminished in infinitum, in the ultimate ratio of a parallelogram to its correspondent one, or which is equal (Hyp.), of one of the parts to its correspondent onc.

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