Chav. VIII.
883
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Fig. 1186.
Fig. 1187.
To reduce rectilineal Figures to one which shall be similarto a given Figure, as those of the trapezium and pentagonto a figure similar to Ma6N: reduce the trapezium andpentagon to triangles, and the figure oiNM to a triangleMaR: reduce the two triangles to the same height, asMaR; draw a triangle equal to these three triangles, bysetting off the bases from a to b, b to c, and c to d, andelevate the perpendicular ce at the point c, of the height,of the triangle MaR; then draw from the point e the linesea and ed, which with the right line ad will form atriangle equal to the three: then reduce the triangleeda to a rectangle fgda, and remark where fg cuts ce inh, and prolong af and dg: take the baseba, and set off on fg, from h to i, anddraw ci until it cuts dg prolonged ink; draw a line parallel to fg through k,until it cuts af prolonged in l; from thepoint l draw lc, remark where it cutsfg in in : draw a perpendicular n m tofg, through the point m, and at *, theperpendicular oip.
Find a mean proportional between thelength be and cp, by bisecting bp in q,and from q as a centre, with the radius qbdescribe a semicircle, and remark where itcuts the perpendicular ce in r,- the length cr will be a mean proportional between be andcp: set off this mean proportional cr from z toe; divide the figure Ma&N into twotriangles by means of the diagonal £»M, and draw on the line CZ , at the point Z, anangle CZs, equal to 6aM; and at the point C, an angle CZs, equal to a&M, and remarkwhere the two lines C s and C t intersect at u ; from this point u draw the angle C ux, equalto 6Ma, and at the point C an angle equal to M6N; the figure uZCz will be equal tothe two triangles, and similar to the figure a&NM.
The above principles are drawn from Prop. 1. of Euclid’s Sixth Book, wherein he statesthat triangles and parallelograms of the same altitude are one to another as their bases, andhe observes in the cor. attached, “ Let their figures be placed so as to have their bases inthe same straight line, and having drawn e
perpendiculars from the vertices of thetriangles to the bases, the straight line whichjoins the vertices is parallel to that inwhich their bases are, because the perpen-diculars are both equal and parallel to oneanother.
To reduce several Circles to a single one, asthe superficies of the two circles A and Bto one : draw the diameter C D and E F;draw a line GH, on which set off CDfrom G to I, and elevate a perpendicularfrom this point I, on which set off theother diameter E F from I to L: drawG L; bisect it in M: from M as a centrewith the radius M L, describe the circleGNLI, which will be equal to the twogiven circles A and B.
To draw a Square the Double or Quadruple of another. — To doublethe square A B C D, prolong the side D E toward the place youwish the square extended : from D as a centre, with the radius D E,describe an arc B E, cutting D C prolonged in F : on D F constructa square GHFD, which will be double the square A B C D.
Vitruvius has given us the method practised in his time fordoubling the square: that valuable author observes, “ If there be anarea or field whose form is a square, and it is required to set outanother field, whose form is also to be square, but double its area, asthis cannot be accomplished by any numbers or multiplication, itmay be found exactly by drawing lines for the purpose, and thedemonstration is as follows: — A square plot of ground, 10 feet
Fig. 1189.
Fig. 1190.
Fig. 1191.
long by 10 feet wide contains 100 feet; if we have to double this, that is, to set out a plotalso square, winch shall contain 200 feet, we must find the length of a side of this
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