QUEST. 25. In the quadrangular field ABCD, if AD be 46.8, DC 19.5, the perpendicular BE 44-8 chains, and D a right angle; alfo if AB be to вс as 8 to 5; it is required to find the content of the trapezium ABCD, and the length of the fides AB, BC. Draw AD 46.8, and DC 19.5 making a right angle D with it; join A, C; take AF: FC :: 8: 5, or AF AC 8:13; make AGAF, and FH parallel to it and to FC; draw GHI meeting AC produced in 1; with the center and radius IF, defcribe an arc meeting in в, a line drawn parallel to, and at the diftance of, the perpendicular 44.8 from AD; and E will be the other point of the figure required.* Calculation. Firft, ACAD2 + DC2 = 507, and, conftruction, 13: 8: AC: 312 = AF; by the hence FC Becaufe, by fimilar triangles, AI: AG or AF::F1: FH or FC, we have, by divifion, AL: FI :: FI: CI, or joining 1B, AI : BI :: BI CI; but the 1, included by thefe proportional lines, is common to both the triangles ABI, CBI, therefore thofe triangles are fimilar, and confequently AB: BC; AI; BI or FI: AG & гH:: 8:5. FCACAF 19.5. Then, by fimilar triangles, 117 (AGFH or AF-FC): 31.2 (= AF or AI -FI) 195 (= FH or FC) 52 FI or BI; whence AI or AF+FI=83.2. Then, by fimilar triangles again, (drawing IK perpendicular to AD produced) as AC 50.7: AI 83.2:: AD 46.8 76.8 = AK, DC 195 32 = KI, whence, (drawing IL parallel to KE), LB OF BE-KI 2 12.8, and EK or LIBI-BL 504; and hence AEAK-EK 26'4, and ABA+EB2 52; alfo 8:5: AB 52: 325 But AEX EB And EDX (BE+ CD) Their fum is BC. 591.36 triang. ABE, 1247.22 fq. chains or 124 ac. 2 г. 35'52 p. the area of the quadrangle ABCD required. QUEST. 26. Given the three fides AB 13, AC 14, and BC 15, of a triangle ABC, divided into three equal parts by the two lines FG, ED, both parallel to AC; it is required to find the areas of the two fegments MHN, KIL, and zone LMNK, into which the infcribed circle is cut by thofe lines. B 1 First, 21 × 8 × 7×6=√72×32×42 = 7×3×4 84 the area of the triangle ABC. I 2 2 Then 847 (AC) 12 is the perpendicular BQ; and 8421 (AB+ AC + BC) = 4 is the radius, or 8 is the diameter H1 of the infcribed circle. Now, the triangles BFG, BED, BAC, are fimilar; but fimilar triangles are as the fquares of their like dimenfions; alfo thofe triangles are to one another as the numbers 1, 2, 3, refpectively; therefore Hence S√1 : 12√3 = 4√3 = BO, √3 BQ 12: [ √2 : 12√3 = 4√6 = BP; BQ-BP12—4√/6=PQ=the verfed fine IR, and BQ-BO12—4√/3=0Q= the verfed fine is; and IH-IS413-4 the versed finesн. Then, dividing the verfed fines HS, IR, by the diameter, we have 413–4 — √3 √3-1 2 =366025404, =275255128; the correfponding tabular verfed fines; to which, in the table of circular fegments, belong the areas •26034449, and 17577983, whose fum 43612432, taken from 78539816, leaves 34927384, the area correfponding to the zone. Then each of these being multiplied by 64, the fquare of the diameter, gives the feveral areas following, viz. = •26034449 16·66204736 the fegment LIK, 64x17577983=1124990912=thefegmentMHN, *34927384=22.35352576 the zone MLKN, PART PART III. MENSURATİON SOLID S. 1. A GENERAL DEFINITIONS. Solid, or body, is a figure extended in every direction. It is commonly faid to confift of length, breadth, and thickness, which are three of its extenfions; of which the direction of each is perpendicular to those of the other two. 2. The measure of a folid is called its folidity, capacity, or content. 3. By the menfuration of folids then are determined the spaces included by contiguous furfaces; and the fum of the measures of thefe including furfaces, is the furface or fuperficies of the body. 4. Solids are measured by cubes, whofe fides are inches, feet, yards, or any other affigned quantity; and hence the folidity of a body is faid to be fo many cubic inches, feet, yards, &c, as will fill its capacity or fpace, or another of an equal magnitude 5. The leaft folid meafure is the cubic inch, other cubes being taken from it according to the proportion in the following table. Table OF PRISMS, PYRAMIDS, AND THE SPHERE, WITH THE PARTS INTO WHICH SOME OF THEM MAY I. BE CUT BY PLANES. Prif is a folid, or A body, whofe ends are any plane figures, which are equal and fimilar; and its fides are parallelograms. A prifin is called a triangular prifin, when its ends are triangles; a fquare prifm, when its ends are fquares; a pentagonal prifin, when its ends are pentagons; and fo on. 2. A cube is a fquare prifm, having fix fides, which are all fquares. It is like a die, having its fides perpendicular to one another. 3. A parallelopipedon is a folid having fix rectangular fides, every oppofite pair of which are equal and parallel. 4. A cylinder is a round prifm; having circles for its ends. 4 00 5. A |