. 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 . and on the oppofite fide, the fur theft up the river. Let b zz breadth AD or AB, and let b reprefentthe iwimmers motion, and d zz BC the currentsmotion, AE zz x9 EB zzy. Then the fwimmergoing off in direction AB, by the compound mo-tion, goes in the line AC, and lands at H, fo that HD is a minimum, by fimilar triangles, AE (#) : EC (d â v) : : AD (b) : DH zz â x~d~^~y zz x d
. 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 . and on the oppofite fide, the fur theft up the river. Let b zz breadth AD or AB, and let b reprefentthe iwimmers motion, and d zz BC the currentsmotion, AE zz x9 EB zzy. Then the fwimmergoing off in direction AB, by the compound mo-tion, goes in the line AC, and lands at H, fo that HD is a minimum, by fimilar triangles, AE (#) : EC (d â v) : : AD (b) : DH zz â x~d~^~y zz x dây . â xyâdâyXXmm. and zz mm. whence £ ill- X XX zz o, and â xy zz d ây X x, but xx + yy zz bb, X DC XXX and xx zz âyy, or ây zz â; therefore - y y zz dx âyx9 and xx zz dy âyy zz bb âyy9 there- bbfore dy zz bb% and y zz â-. d Cor. i. The direclion AC will touch the circleGCH, equal to the circle KBD ; for all the lines GK,CB, cb, &c, are equal -9 d being greater than b. Cor. 2. If b is equal or greater than d, the fwim*mers direction is extremely near AK. Cor. 3. I ID zz s/ddâbb. Cor. 4. If the time of paffmg through AB be zz t j dtthe time of arriving at H will be x/ddâbb For. jMisodlani&s. j?;i. / \i Art, XVIII. PROBLEMS. \69 / ^4 \ i Fig For at (v/^ â -7w : ** : *b : tmie fought. & Sc H O L. The river will be crofted in the lead time, bymoving in the perpendicular direction AD. P R O B. XXXI. Let CD be an exponential curvey wh&fe equation is nuy zr xx, putting AB zz xy BD zz y. To find the area,and the leafi ordinate. i. Let X â: hyp. log. »,Yz hyp. log. v. Thenfince y zz x*9 therefore Y zz xX ; and by the na-ture of logarithms (fee Schol. 2. Prob. II. Sect. II. Yz Y* Y+ Fluxions,) y zz 1 + Y + h -7- -\ &c. 2 6 24 xzXz #>X?zz 1 + xX + + , &c. and the fluxion 2 3of the area yx or x*x zz x + Xxx + < -+⢠Sfi XWx XWx 0 ⢠: . a â 1 f &c. And the fluent zz x + 6 24 xxX x*Xz #00 fl + -â- -\ &c. â s. And proceed- 2 6 24 xz #;X ing by R
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