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Cor. 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 resolutions abundantly confirmed from mechanicks.

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.

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If in any figure BacE 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, &c. be inscribed, contained under equal bases AB, BC, CD, doc. and sides Bb, Cc, Dd, &c. parallel to the side sa of the figure, and the parallelograms a Kbl,, Mon, &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 AKb Lidl, 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 KI, 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.




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

breadths AB, BC, CD, foc. 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.


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 two figures are to each other, in the same ratio.

For as the parallelograms in one figure are to those in the other, each to cach, 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, parallelo

, grams 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 one.

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All corresponding sides of similar figures, as well curvilineal, as

rectilineal, are proportional ; and the areas are in a duplicate ratio of the corresponding sides.


If any arch (AB) given by position, be subtended by a chord (AB),

and, in any point (A) in the middle of the continual curvature, be touched by a right line ( AD), produced both ways, and its extreme points (A and B) approach each other and come together; the angle (BAD) contained by the chord and tangent will be diminished in infinitum, and ultimately vanish.

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The same things being supposed ; the ultimate ratio of the arch,

chord, and tangent to each other, is the ratio of equality.

For while the point B, see fig. to prec. lemma, approaches to the point A, let the right lines AB and AD be always understood to be produced to distant points b and d, and to the secant or cutting line BD, let bd be drawn parallel, and let the arch Ab be always similar to the arch AB. And, the points A and B coinciding, the angle dAb, by the preceding lemma, vanishes, and therefore the right lines Ab and Ad, which are always finite, and the intermediate arch Ab, coincide and are therefore equal. Whence also the right lines AB and AD, always proportional

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to the right lines Ab and Ad (4. 6 Eu.); and the intermediate arch AB, have to each other, an ultimate ratio of equality.

Cor. 1. Whence if through B, a right line BF be drawn paral

EV lel to the tangent, always cutting any right line Af passing thro' A, in F; this right line BF, has F


B ultimately to the evanescent arch AB, a ratio of equality ; because, the parallelogram AFBD being completed, it has always a ratio of equality to AD.

Cor. 2. And if through B and A, there be drawn more right lines BE, BD, AG and AF, cutting the tangent AD and its parallel BF; the ultimate ratio of all the abscissas or right lines cut off AD, AE, BF, BG, and of the chord and arch AB, each other, is the ratio of equality,

Cor. 3. And therefore all these lines, in all reasoning about ultimate ratios, may be used for each other.



If the two right lines AR and BR, see the figure to lemma 6, with

the arch AB, the chord B, and the tangent 4D, from three triangles ARB, AB and ARD, and the points A and B approach, and come together ; the ultimate form of the evanescent triangles, is that of similitude, and the ultimate ratio, that of equality.

For while the point B approaches to the point A, let AB, AD and AR be always understood to be produced to distant points b, d and r, the right line rbd to be drawn parallel to RD, and let the arch Ab be always similar to the arch AB. And the points A and B coinciding, the angle dAb, by lemma 6, vanishes, and therefore the three triangles rAb, rAb and rAd, which are always finite, coincide, and are therefore similar and equal. Whence also the triangles RAB, RAB and RAD, which arc always similar and proportional to these, become ultimately similar and equal to each other.

Cor. And hence, these triangles, in all reasoning about ultimate ratios, may be used for each other.

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