EXAM. 11. To find the greatest rectangle that can be inscribed in a given right-angled triangle.

EXAM. 12. To find the greatest rectangle that can be inscribed in the quadrant of a given circle.

EXAM. 13. To find the least right-angled triangle that can circumscribe the quadrant of a given circle.

ΕΧΑΜ. 14. To find the greatest rectangle inscribed in, and the least isosceles triangle circumscribed about, a given semi-ellipse.

EXAM. 15. To determine the same for a given parabola. EXAM. 16. To determine the same for a given hyperbola. EXAM. 17. To inscribe the greatest cylinder in a given cone; or to cut the greatest cylinder out of a given cone.

EXAM. 18. To determine the dimensions of a rectangular cistern, capable of containing a given quantity a of water, so as to be lined with lead at the least possible expense.

EXAM. 19. Required the dimensions of a cylindrical tankard, to hold one quart of ale measure, that can be made of the least possible quantity of silver, of a given thickness.

EXAM. 20. The cut the greatest parabola from a given

cone.

EXAM. 21. To cut the greatest ellipse from a given cone. ΕΧΑΜ. 22. To find the value of x when at is a minimum.

THE METHOD OF TANGENTS; OR OF DRAW. ING TANGENTS TO CURVES.

96. THE Method of Tangents, is a method of determining the quantity of the tangent and subtangent of any algebraic curve; the equation of the curve being given. Or, vice versa, the nature of the curve, from the tangent given.

If AE be any curve, and E be any point in it, to which it is required to draw a tangent TE. Draw the ordinate ED: then if we can determine the subtangent TD, limited between the ordinate and tangent, in the axis produced, by joining the points T, E, the line TE will be the tangent sought.

E

الا

e

a

TAX Dd C

97. Let dae be another ordinate, indefinitely near to DE, meeting the curve, or tangent produced in e; and let Ea be parallel to the axis ad. Then is the elementary triangle wea similar to the triangle TDE; and

which is therefore the general value of the subtangent sought; where x is the absciss AD, and y the ordinate DE.

Hence we have this general rule.

GENERAL RULE.

98. By means of the given equation of the curve, when

put into fluxions, find the value of either i or ý, or of

yż y

which value substitute for it in the expression DT = and, when reduced to its simplest terms, it will be the value of the subtangent sought.

EXAMPLES.

EXAM. 1. Let the proposed curve be that which is defined, or expressed, by the equation ax2 + xy2 - y3 = 0.

Here the fluxion of the equation of the curve is 2axi + y2i + 2xyj-3y2ý = 0; then, by transposition, 2axi + y2i = 3y2y - 2xyy; and hence, by division,

which is the value of the subtangent TD sought.

EXAM. 2. To draw a tangent to a circle; the equation of which is ax-r2 = y; where x is the absciss, y the ordinate, and a the diameter.

EXAM. 3. To draw a tangent to a parabola; its equation being px = y2; where p denotes the parameter of the axis. EXAM. 4. To draw a tangent to an ellipse; its equation being c2(ax - x2) = a2y2; where a and c are the two axes. EXAM. 5. To draw a tangent to an hyperbola; its equation being c2 (ax + x2) = a2y; where a and care the two EXAM. 6. To draw a tangent to the hyperbola referred to the asymptote as an axis; its equation being xy = a2; where a2 denotes the rectangle of the absciss and ordinate answering to the vertex of the curve.

By slight and obvious extensions of the same principles, tangents may be drawn to spirals, and asymptotes may be drawn to such curves as admit of them.

OF RECTIFICATIONS; OR, TO FIND THE
LENGTHS OF CURVE LINES.

99. RECTIFICATION, is the finding the length of a curve line, or finding a right line equal to a proposed curve.

By art. 10 it appears, that the elementary triangle Eae, formed by the increments of the absciss, ordinate, and curve, is a right-angled triangle, of which the increment of the curve is the hypothenuse; and therefore the square of the latter is equal to the sum

of the squares of the two former; that is, Ee2 = a2 + ae”. Or, substituting, for the increments, their proportional fluxions, it is żż = iż + ýý, or z = √(i+j); where z denotes any curve line AE, x its absciss AD, and y its ordinate DE. Hence this rule.

RULE.

100. From the given equation of the curve put into fluxions, find the value of i or y, which value substitute instead of it in the equation z = √(x2+y2); then the fluents, being taken, will give the value of z, or the length of the curve, in terms of the absciss or ordinate.

EXAMPLES.

EXAM. 1. To find the length of the arc of a circle, in terms both of the sine, versed sine, tangent, and secant.

The equation of the circle may be expressed in terms of the radius, and either the sine, or the versed sine, or tangent, or secant, &c. of an arc. Let therefore the radius of the circle be ca or CE = r, the versed sine AD (of the arc AE)=x, the right sine DE = y, the tangent TE = t, and the secant cr = s; then, by the nature of the circle, there arise these equations, viz.

Then, by means of the fluxions of these equations, with

the general fluxional equation z2 = i + y2, are obtained the

following fluxional forms, for the fluxion of the curve; the

fluent of any one of which will be the curve itself; viz.

Hence the value of the curve, from the fluent of each of

these, expressed in series, gives the four following forms, in

series, viz. putting d = 2r the diameter, the curve is

Now, it is evident, that the simplest of these series, is the third in order, or that which is expressed in terms of the tangent. That form will therefore be the fittest to calculate an example by in numbers. And for this purpose it will be convenient to assume some arc whose tangent, or at least the square of it, is known to be some small simple number. Now, the arc of 45 degrees, it is known, has its tangent equal to the radius; and therefore, taking the radius r = 1, and consequently the tangent of 45°, or t, = 1 also, in this case the arc of 45° to the radius 1, or the arc of the

* These formulæ are, obviously, analogous to those given in art. 30, p. 312, and are so many forms of fluxions whose fluents become known.

is a circular arc V(2rx-x2)' The fluent of an expression

Thus the fluent of an expression, such as whose radius is = rand versed sine = x. such as is a circular arc whose radius is = r and tangent = t : 72+13 and so of the rest.

r2t

Conversely, the same formulæ, or those just referred to, serve to assign the relative magnitudes of the differences in any parts of a table of

natural sines, of natural tangents, &c. Thust =

r2+t2.
r2

= z X sec.2 of

arc to tan. t, consequently, the tabular differences of the tangents vary as the squares of the secants. Hence, those differences, at 0o, at 45°, and at 60°, are as 13, (2)2, and 22, or as 1, 2, and 4. This suggests an application of these formule which will often be found useful.

quadrant to the diameter 1, will be equal to the infinite series 1-+-+ +- &c.

But as this series converges very slowly, it will be proper to take some smaller arc, that the series may converge faster; such as the arc of 30 degrees, the tangent of which is = √, or its square t2 = 1: which being substituted in the series, the length of the arc of 30° comes out

pute these terms in decimal numbers, after the first, the succeeding terms will be found by dividing, always by 3, and these quotients again by the absolute numbers, 3, 5, 7, 9, &c.; and lastly, adding every other term together, into two sums, the one the sum of the positive terms, and the other the sum of the negative ones; then lastly, the one sum taken from the other, leaves the length of the arc of 30 degrees; which being the 12th part of the whole circumference when the radius is 1, or the 6th part when the diameter is 1, consequently 6 times that are will be the length of the whole circumference to the diameter 1. Therefore, multiplying the first term by 6, the product is ✓ 12 = 3·4641016; and hence the operation, true to 7 places of decimals, will be conveniently made as follows: