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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.
For if that angle should not vanish, the arch AB would contain, with che tangent AD, an angle equal to a rectilineal one, and therefore the curvature at the point A would not be continual, contrary to the supposition.
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 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 parallel 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, to 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 AB, and the tangent 4D, from three triangles ARB, AWB and ARD, and the points 1 and B approach and come together ; the ultimate foron 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 angledAb, 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 are 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.
If a right line AE, and a curve ABC, given by position, cut each
other in a given angle at A, and to that right line, in another given angle, BD and CE be ordinately applied, meeting the curve in B and C, and the points B and C approach and come together to the point A ; the areas of the triangles ABD and ACE are to each other ultimately, in a duplicate ratio of the sides.
For while the points B and
e C approach to the point A, let the right line AD be understood to be produced to distant
ď points d and e, so that Ad and Ae may be proportional to AD and AE, and let the ordinates db and ec be drawn par
F allel to DB and EC, which may meet the right lines AB and AC produced in b and c. Let there be understood to be
A drawn, both the curve Abc similar to ABC, and the right line AG, which may touch both curves in A, and cut the ordinates DB, EC, db and ec in F, G, fand g. And, the length Ae remaining the same, let the points B and C come together to the point A, and, the angle cAg vanishing, the curvilineal areas Abd and Ace will coincide with the rectilineal ones Afd and Age, and therefore, by lemma 5, will be in a duplicate ratio of the sides Ad and Ae (19. 6 Eu.); but to these areas, the areas ABD and ACE, and to these sides, the sides AD and AE are always proportional; therefore the areas ABD and ACE are to each other ultimately in a duplicate ratio of the sides AD and AE.
The spaces, which a body describes, by any finite force urging it,
whether that force be determined and immutable, or be continually increased or continually diminished, are, in the very beginning of the motion, in a duplicate ratio of the times.
Let the times be represented by the right lines AD and AE, see figure to the preceding lemma, and the velocities generated, by the ordinates DB and EC ; the spaces described by these velocities, are as the areas ABD and ACE described by these ordinates, or, which is equal by the precediug lemma, in a duplicate ratio of the times AD and AE.
The evanescent subtense of the angle of contact, in all curves hav
ing a finite curvature at the point of contact, is ultimately in a duplicate ratio of the subtense of the conterminous arch.
Case 1. Let that arch be AB, its tangent A
D AD, the subtense of the angle of contact BD,
16 and the subtense or chord of the contermi
B nous arch or the arch having the same ex- C tremes, the right line AB; and first, let the subtense BD of the angle of contact be perpendicular to the tangent AD. To the subtense AB and tangent AD, erect the perpendiculars BG and AG, meeting each other in G, and let the points D, B and G approach J to the points d, b and g, and let j be the intersection of the right lines BG and AG,
G made ultimately, when the points D and B come to A. Itis manifest that the distance GJ may be less than any given right line. But, because of the right angled triangle ABG, the square of AB is equal to the rectangle GAC (Cor. 1. 8 6 and 17. 6 Eu.), or, AC and BD being equal, to the rectangle under AG and BD; for the same reason, the square of the right line Ab is equal to the rectangle under Ag and bd; therefore the square of AB is to the square of Ab, in a ratio compounded of the ratios of AG to Ag and of BD to bd (23. 6 Eu). But because GJ may be assumed less than any given length, the ratio of AG to Ag may be such, as to differ from the ratio of equality less than by any given difference, and therefore the ratio of the square of AB to the square of Ab may be such, as to differ from the ratio of BD to bd less than by any given difference; and therefore, by lemma 1, the ultimate ratio of the square of AB to the square of Ab is equal to the ultimate ratio of bD to Bd, and so bD is to bd ultimately in a duplicate ratio of the subtenses AB and Ab (20. 6 Eu).
Case 2. Let now BD be inclined to AD'in any given angle, and the ultimate ratio of BD to bd will still be the same as before (4. 6 Eu.), and therefore in a duplicate ratio of the subtenses AB and Ab.
Cnse 3. And though the angle D should not be given, but be formed by the right line BD converging to a given point, or by any other law; yet the angles D and d, constituted by a common law, always tend to equality, and approach nearer to each other, than by any given difference, and are therefore ultimately equal, by lemma 1 ; and therefore the lines BD and bd are to each other, in the same ratio as before.
Cor. 1. Since the tangents AD and Ad, the arches AB and Ab, and their sines BC and bc, become ultimately equal to the chords AB and Ab; their squares are ultimately, as the subtenses BD and bd.
Cor. 2. Their squares are also ultimately as the sagittas or versed sines of the arches, bisecting the chords, and tending to a given point. For these sagittas are as the subtenses BD and bd.
Cor. 3. And therefore the sagitta is in a duplicate ratio of the time, in which a body with a given velocity describes an arch; the arch described with a given velocity being as the time.
Cor. 4. The rectilineal triangles ADB and Adb are ultimately in a triplicate ratio of the sides AD and Ad, and in a sesquiplicate of the sides DB and db, as being in a ratio, compounded of the ratios of the sides AD and DB to Ad and db (Cor. 1. 23. 6 Eu). So also the rectilineal triangles ABC and Abc are to each other ultimately in a triplicate ratio of the sides BC and bc (See Def. 14. 5 Eu).
Cor. 5. And because DB and db are ultimately parallel, and in a duplicate ratio of AD and Ad, and therefore AC and Ac ultimately in a duplicate ratio of BC and bc, which is the nature of the parabola (Cor. 2. 40. 1 Sup.); the ultimate curvilineal areas ADB and Adb are two thirds parts of the rectilineal triangles ADB and Adb (Cor. 81. 1 Sup.); and therefore the ultimate segments AB and Ab third parts of the same triangles. And therefore these areas and segments are in a triplicate ratio both of the tangents AD and Ad, and of the chords of the arches AB and Ab.