Cyclomathesis : or, An easy introduction to the several branches of the mathematics; being principally designed for the instruction of young students, before they enter upon the more abtruse and difficult parts thereof . AXIMA W B. It Fig. the area of the poligon AECDE infcribed in afemicircle. PROB, CCXXVIII. 1^2.^0 find a line, which with three given lines, will con*tain the greatefi area poffible* It is plain the line fought is the diameter of thefemicircle in which the three given lines are in-fcribed. Let ABCD, be the quadrangle, draw the dia-gonals AC, BD, on which let fall the perpendi-
Cyclomathesis : or, An easy introduction to the several branches of the mathematics; being principally designed for the instruction of young students, before they enter upon the more abtruse and difficult parts thereof . AXIMA W B. It Fig. the area of the poligon AECDE infcribed in afemicircle. PROB, CCXXVIII. 1^2.^0 find a line, which with three given lines, will con*tain the greatefi area poffible* It is plain the line fought is the diameter of thefemicircle in which the three given lines are in-fcribed. Let ABCD, be the quadrangle, draw the dia-gonals AC, BD, on which let fall the perpendi-culars CP, BF. Let ABzzb, BC=r and ACrzVj/y—dd. But(Geom. IV. 28.) CP zz -, and BF= t. There- y y fore h + — v/jy — bb ~ 2 area ABCD, and « + — Vyy~dd zz 2 area zr 0 -| vyy-bb9 and by+cd X \/yy—b£> zz dy+bc y/yy—dd* and fquar-ing and multiplying, bby+ -+- ibedy* 4- fft%y—b*yy—2b*cdy—bbcoddzzddy* + 2bcdy* 4- bbccyy— d+yy— 2bcdy — bbcedd. And reducing bby^— b* y + 2bcd* zz o.— dd — bbec — 2blcd. -f- cedd4- d*And dividing by bb — ddyi—hby zz 2lcdy and y being known, the—cc—ddarqa is known from the foregoing fteps. PRO */.,-^ Sea. xiv. minima;, 5i3 [fig* P R O B. CCXXIX. SP is perpendicular to PM, and there is given SP, •, and drazving NL, fo that the angle LDMmay be equal to SCP 5 to find CD, a maximum* Draw NA perpendicular to CD, then CA = AD,and CA is a maximum. Put SNz^, SP zz d,SCzzy, then CN=i~jf, CP = Vjf^dd. Thenby fimilar triangles, y : s/yy—dd : : £-—y : CA zz £—y _ yy—dd vyy—dd zz max. and b—tyv X — = max. = b^\z — — X b^yf- Increafe j by a very fmall quantity <?, then b-y-e\% zz £~)i* — 2e X b—y. Alio y -f e\x zz yy + 2j£, and by divifion dd dd idde „ * • dd — = r— Whence b—y\ —— X yy -f- 2j 1 = — X b—y + j\ and multi-plying by y\ we have j*i z: ^ — ddy + di% or j zz bdd, and j zz Vbdd* li prob; 514 MAXIMA and B. II, Fig. P R
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