(a−d√1—s2)2 +d2 s2=(b+c√1−s2)+c2s2, and expanding, a2 —2ad√ 1−s2 +d2 —d2 s2+d2 s2=b2+2bc √ 1—s2 +c2-c2s2+c2s2, and cancelling terms, a2-2ad1-82+ d2 = b2+2bc√1—s2+c2. Transposing and changing signs, (2ad+2bc) 1-s2=-(b2+c2-a2-d2). Dividing by 2ad +2bc, √1-82 (b2+c2-a2-d2) (2ad +2bc), and squaring each side, 1-s2= (b2 +c2 —a2 —d2)2÷(2ad+2 bc)2. By transposing and changing signs, s2=1-(b2+c2 -a2-d2)2(2ad+2bc)2. Now 1 any number divided by itself; therefore, s2 _ (2ad+2bc)2 _ (b+c2-a2-d2) (2ad+2bc)2 (2ad+2bc)2- (b2+c2 — a2 d2) (2ad+2bc)2

root, s=

(2ad+2bc)2

and by extracting the sq.

✓ (2ad+2bc)2 — (b2 +c2 —a2 —d2)2. Substituting

2ad+2bc

this value of s in RULE I, and we have area

✓ (2ad+2bc)-(b2+c2-a2-d2)2

2x (ad+bc)

(2a+2bc)2 - (b2+c2—a2—d2)2

2×2× (ad+bc)

=

ad+bc

2

by cancelling equal

and

factors both from the numerator and the denominator, √(2ad+2bc)2 —(b2+c2—a2—d2)2. By resolving the radical part as in Trig. 221, (b+c)2- (a-d)2=the sum, (a+d)2- (bc) the diff. By resolving again, (b+c+a-d)=(a+b+c-d)=the sum of the first : (b+c-a+d)=(b+c+d-a)=the difference of the first: (a+d+b-c)=(a+b+d-c)=the sum of the second: (a+d-b+c)=(a+d+c-b)=the difference of the second: and, these four factors when multiplied together equal the quantity under the radical sign in the preceding equation; hence by substitution, area=√(a+b+c−d)x(b+c+d√a)×(a+b+d—c)×(a+d+c—b) If we put h=(a+b+ c+d)= the sum of the sides, a+b+c¬d=2h−2d, b+c+ d-a-2h-2a, a+b+d−c=2h-2c, and a+d+c−b=2h -26; then the area=2(h-a)×2(h-b)×2(h—c)×2

√(h−d)=√16×(h—a)×(h—b)×(h−c)×(h−d)=by re

moving a factor from under the radical sign 4×1√(h -a √(h—b)x(h-c)× (h-d)=√(h-a) × (h—b) x (h-c) x ✓(h-d)=the area.

PROBLEM V. To find the area of a regular polygon.

Example 1. 72.426÷2=36,213, and 36.213×60×2 17382.240 the area.

=

Example 2. 170.7867-35.3938 which multiplied by 40 ×10=16281.148 the area.

Example 1. Art. 16. By the formula,(Fig. 7.) R ; BP::cot BCP CP. Substitut. numbers, 1:19::cot 30°: CP; and, by the tables, 10.0000000; 1.2787536::10.2385606 1.51 73142=32.908204, which divided by 2, this quotient multiplied by 38, and the product multiplied by 6=3751.535256 the answer.

Example 2. By Trig. 122, as in the preceding example, R 362 cot 360-20 CP; and, by the tables, 10.00000 00: 1.4913617::10.4882240; 1.9795857 = 95.40819780, which multiplied into 31, and into 10-29576.541318=the

answer.

From the preceding proportion, perpendiculars and areas may be easily formed, for a series of polygons, of which each side is a unit, by which may be calculated the area of any other regular polygon, of the same number of sides. with one of these; for the areas of similar polygons (Euc. VI. 20.) are as the squares of their homolagous sides.

Example 1. Art. 17. 1022 × 7.6942088=10404 × 7.6942 088-80050.5483552=the area.

Example 2. 872 × 11.1961524-7569 × 11.1961524=847 43.6774156 the area.

=

THE QUADRATURE OF THE CIRCLE AND ITS PARTS.

The squaring of the circle is a problem which has exercised the attention of distinguished mathematicians ever since the early history of the science. It was studied by Archimides more than 2000 years ago; and, although the result of his efforts have been essentially the same as those who have succeeded him, still almost every one fond of mathematical research, vainly indulges the idea that he can ac complish what many of gigantic powers have been compelled to acknowledge themselves incompetent to perform. Le. gendre says of this problem, that it is now degraded to the rank of those idle questions, with which no one possessing the slightest tincture of geometrical science, will occupy, any portion of his time. And yet, since the present treatise has been in progress, a Sophomore of high rank in his class, mentioned to the writer with much seeming confidence, that he believed he could solve it.

The problem of the quadrature of the circle consists in finding a square equal in surface to a circle, the radius of which is known. Now it has been proved (Sup. Euc. I, 5.) that a circle is equivalent to the rectangle contained by half its circumference, and its radius; and this rectangle may be changed into a square, by finding a mean proportional between its length and breadth. (Euc. VI. 11.) To square the circle, therefore, is to find the circumference when the radius is given; and for effecting this, it is enough to know the ratio of the circumference to its radius, or its diameter. Hitherto, the ratio has never been determined except approximately; but the approximation has been carried so far, that a knowledge of the exact ratio would afford no real advantage whatever beyond that of the approximate ratio. Archimides showed that the ratio of the circumference to the diameter is included between 340 and 319. More accurate values have since been obtained for the same number.

At

length the value of the circumference developed to a certain order of decimals, considering the diameter equal to unity. Some have had patience enough to continue these decimals to the 127th and even to the 140th place. Such an approximation is evidently equivalent to perfect correctness. Two simple methods of obtaining these approximations are exhibited in the 293d and 294th problems of Legendre's geometry. Also, a demonstration that the diameter and the circumference of a circle are incommensurable, may be found in Note IV. to the same. The circumference of a circle whose diameter is 1, is 3.141592, 653589, 793238, 462643, 383279, 502884, 197169, 399375, 105820, 974944, 592307, 816406, 286208, 998628, 034825, 342117, 067982, 148086, 513272, 306647, 093844, 6+ or 7-.

Until the invention of fluxions, and the modern improvements in analysis, most methods of approximation were exceedingly laborious. But ways have since been devised which are expeditious, and often quite simple. These principally consist in finding the sum of a series, in which the length of an arc is expressed in terms of its tangent. It would be interesting to attend particularly to several of these systems, but our limits will not permit any thing further on this subject than the reduction of a few expressions to relieve the student of the drudgery of long multiplications and divisions, which is sometimes imposed on him.

We would add, however, that although a definite quadra. ture of the whole circle has never yet been given, nor of any aliquot part of it; yet, certain other portions of it have been squared. A partial quadrature was given by Hippocrates of Chios, who squared a portion called, from its figure the lune, or lunule; but this quadrature has no dependence on that of the circle. Some modern geometricians have found out the quadrature of any portion of the lune taken at pleasure, independently of the quadrature of the circle; though still subject to a certain restriction, which prevents the quadrature from being perfect, and what the geometri. cians call absolute and indefinite.

By Trig. 223 the cosine of an arc (radius being 1,) of 24376 of the whole circumference=.99999996732, and the sine=.00025566346. By Trig. 228, tan=sin÷cos; that is, tan=.00025566346÷.99999996732.

.99999996732).00025566346 (.00025566347=tan.

199999993464

556634665360

499999983660

566346817000

499999983660

663468333400

599999980392

634683530080

599999980342

346835496880

299999990196

468355066840

399999986928

683559799120

677777777124

57730219960

By Subtracting the sine from the tangent, we find the difference to be only .00000000001, and the difference between the sine and the arc is still less. Taking then .000255663465 for the length of the arc, multiplying by 24576, the denominator of the fraction denoting what part the arc is of the whole circumference, and retaining 8 places of decimals, we have 6.28318531 for the whole circumference.

.000255663465

24576

001533980790

001789644255

001278317325

001022653860

000511326930

6.283185315840, which divided by 2= 3.141592657920-the circumference of a circle whose radius is, and diameter 1. If 3.14159265 be multiplied by 113, the product is 354.99996945+, or 355 very nearly. We obtain, then, the result: Diam. : Circum.::113 : 355.

Note. For those students who wish to gather interesting facts in reference to the almost endless investigations of mathematicians