bolizes approximately a straight line, of which the length is æ, and the breadth, if one may so speak, is dr; but (da)2 represents a square whose side is dæ; and as dæ is an infinitesimal, its square is a point, and as it will require an infinity of such points to make a straight line, and as the coefficient of (da) is not infinity, we must neglect it; that is, in calculating the enlargement of the square due to the enlargement of a side, we take account of the infinitely narrow rectangles which adjoin the sides, but must neglect the small point which is required to complete the square, and which is situated at one of the angles, as at B, and no error is committed by our so doing. Or, if we introduce the idea of motion, the enlargement of the square is due to the moving forwards of the two sides PB and CB, and the rectangles by which the square is increased are the several spaces passed over by the sides, which are the spaces contained between the lines before and after the motion; and as the spaces through which the lines have passed are very small, the lines being considered to be in two immediately successive positions, the small element at в becomes a point, and, as we have not an infinity of such points, the accuracy of our result is not destroyed by neglecting this small quantity; and therefore, again, the increase of the square due to the infinitesimal increase of the side is 2x da.

{a2 + (x + ▲x)2 } #

(x +▲x) { a2 + x2} 1⁄2

x + AX

{a2 + ( x +▲x)2 } + '

{a2 + x2 } }

x { a2 + x2 + 2 x sx + (sx)2 } $ {a2 + x2} + {a2 + (x +▲x)2 } ÷

and expanding the second member of the numerator by the Binomial Theorem, and neglecting the terms involving the square and higher powers of Ar, which will become infinitesimals of an order to be omitted, we have

PRICE, VOL. I.

G

Ay =

=

(x +▲x) { a2 + x2 } 3 — x { (a2 + x2) 3 + x ▲x (a2 + x2)− §}

whence, taking differentials, and omitting dx, where it is added to the finite quantity x, we have

to evaluate e-1, when ▲x becomes an infinitesimal dx;

replacing therefore e-1 by its equivalent in the above equation, and omitting do when added to the finite quantity x, we

.. y +▲y = cos (x +▲x) sin 2 (x + ▲x);

▲y = cos(x + ▲x) sin 2 (x +▲x) — cos x sin 2x

= (cos x cos x - sin x sin ax) (sin 2x cos 2 ax + cos 2x sin 2 ax)

cos x sin 2x; and taking differentials instead of differences, and therefore by reason of Lemma II replacing the sine of an infinitesimal arc by the arc itself, and the cosine by unity, we have

dy = (cosx-da sin x) (sin 2x + 2 dx cos 2 x) — cos x sin 2x, = 2 dx cos x cos 2x dx sin x sin 2x,

dy

-

omitting, as is necessary, the term involving (dx)2. If therefore

= cos x sin 2x,

f'(x) = 2 cos x cos 2x

--

f(x) = sin x sin 2x. The above examples are instances of differentiating from first principles.

Ex. 5. To determine the perimeter and the area of a circle of given radius.

As in this and the two following examples, and indeed throughout the whole work, it is necessary to have a clear notion of the relation between circular and gradual (that is, by degrees) measures of angles, I propose once for all to say a few words on the subject.

In Euclid, Book VI, Prop. XXXIII, it is proved that in the same circle, or in equal circles, angles at the centre have the same ratio which the arcs on which they stand have to each other: hence we conclude that in the same circle, or in equal circles, the arcs vary as the angles which they subtend at the

centre.

Also let us suppose in two circles of unequal radii, which we will assume to be concentric, although this assumption is unnecessary, two regular polygons of the same number of sides to be inscribed; then the perimeters of these polygons evidently vary as the radii of the circles. Let the number of the sides be infinitely increased, so that each side becomes a straight line of infinitesimal length: the lengths of these sides evidently still vary as the radii; for the infinite increase in the number of these sides, and thereby the infinitesimal diminution of the lengths of the sides, does not change this ratio. But when the

number of the sides is infinite, the polygon becomes a circle, and its perimeter becomes the circumference of the circle; so that by this method we infer that the perimeters of circles vary as their radii. The same relation is also true of any parts of the perimeters; and therefore of two concentric circles the arcs subtending the same angle at the common centre vary as the radii of the circles. If therefore we combine this result with the preceding, we have the following proposition:

The circular arcs subtending different angles with different radii vary conjointly as the angles and the radii of the arcs.

Let s, r, be the symbols representing an arc, a radius, and the subtended angle; these symbols expressing numbers; the length-unit of the radius and are being the same, and the angleunit being at present undetermined. Let k be a coefficient of variation. So that the preceding proposition expressed mathematically is

s = kro.

But k is undetermined; to determine it, let us assume the unitangle to be that, the subtending arc of which is equal to the radius; let the importance of this assumption be noted. Then in the preceding equation = 1, when s=r; and therefore k = 1; and

Now a right-angle is a quantity independent of any arbitrary assumption. In ordinary trigonometry it has been found convenient to divide it into 90 equal parts or angles, each of which is called a degree: this indeed is the definition of a degree.

Here then are two different unit-angles. The latter is much more arbitrary than the former; we might divide a right angle into 100 or into any other number of equal parts, if such a division were more convenient than that into 90 equal parts. But the principle of the former is founded on certain geometrical properties of the circle. Thus then we have, what we will call technically, the unit-angle, and a degree. A question arises, what relation exists between these angles? can the relation be expressed by a fraction? or by any other number? We reply, no; no number commensurable with our scale can express it. We may roughly get a notion of the relation in the following way:

The unit-angle is that of which the subtending arc is equal to the radius: now the chord of 60 degrees is equal to the ra

dius; and the chord is less than the arc: so that the unit-angle is less than 60°; it ultimately results that the unit-angle is 57.29578... degrees. Let us however assume the number which gives the ratio of two right angles to the unit-angle to be represented by so that is the number of times the unit-angle is contained in two right angles. Thus if represents the unitangle,

TX = 180°.

(17)

The numerical value of π (to five places of decimals) is 3.14159..... and the fraction which most conveniently, although of course not

355
113

the number is however incom

correctly, represents it is mensurable, and has actually been calculated within the last four years by Mr. Shanks of Houghton le Spring in the county of Durham, to the astonishing number of 607 places of decimals. Let the reader therefore be careful as to the meaning of the symbol : it is a number, and a number only. By some authors it has been used as equivalent to two right angles or 180°: such an use is incorrect as to form; and although commonly used can be justified only by understanding it to be the coefficient of e, which is the unit-angle; and thus makes the quantity Te to be a concrete angle.

In the same way I would observe, that when the symbol a or ℗ is used for an angle the symbol is a number, and implies that a times or times the unit-angle is taken.

One other remark must be made; the sine, cosine, tangent, &c. are affections of angles, and not of circular arcs: but since by the preceding equation s=0, if r=1; the trigonometrical functions become affections of arcs of a circle, if the radius is unity and thus if x is an arc, sin x, tan a, &c. are intelligible.

Now to return to our problem: let ACD, fig. 4, be the circle, of which let the radius be a; take o the centre, and let the angle AOC be the nth part of four right angles, and ac be the side of a regular polygon of n sides inscribed in the circle; and make a construction such as is represented in the figure. Then if 2 is the symbol for four right angles,

..

the perimeter of the polygon = n.Ac=2n.AB = 2 na sin;

n