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July 21, 2024, 1:33 pm

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ABxAF: abx af:: A af:: A B3: Aab. 101 Draw the radius BO. Page 1 LOO ffIS7S SERIES OF SCHOOL AND COLLEGE THE Course of Mathematics by Professor Loomis has now been for several years before the Public, and has received the general approbation of Teachers throughout the country. In a right-angled triangle, the square on either of the two sides containing the right angle, is equal to the rectangle contained by the sum and difference of the other sides. When three straight lines, as AB, CD, EF, are perpendicular to each other, each of these lines is perpendicular to the plane of the other two, and the three planes are perpendicular to each other. Hence the remaining parts of the triangle ABC, will be B E equal to the remaining parts of the triangle DEF; that is, the side A D will be equal to DF, BC to EF, and the angle ACB to the angle DFE Therefore, if two trianales, &c. Page 160 160 GEOMETRY. If such can not be found, draw other lines, parallel or perpendicular, as the case may require; join given points or points assumed in the solution, and describe circles if necessary; and then proceed to trace the dependence of the assumed solution on some theorem or problem in Geometry.

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Let AB, CD be the two parallel _ straight lines included between two _ 7 parallel planes MN, PQ; then will AB -- be equal to CD. The square of any diameter, is to the square of its conjugate. DF; and let planes' pass through these lines and the vertex A; they will divide the polygonal pyramid? AC is any diameter, and BD its parameter; then is BD A equal to four times AF. To discover whether a surface is plane, we apply a straight line in different directions to this surface, and see if it touches throughout its whole extent. Hence F'K-FK

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That the convex surface of a frustum of a pyramid is equal to the product of its slant height, by the perimeter of a section at equal distances between its two bases; hence the convex surface of a frustum of a cone is equal to the product oj its side, by the circumference of a section at equal distances between tile two bases tiI. Cylinders of the same altitude, are to each ot aer as their bases; and cylinders of the same base, are to each other as their altitudes. And its lateral faces AF, BG, CH, DE are rectangles. Because the alternate angles ABE, ECD o are equal (Prop.

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Page II Entered, according to Act of Congress, in the year 1858, b3 ELIAS LooMIs, In the Clerk's Office of the Southern District of New York. Again, the angle BGF is equal to the angle AGE (Prop V. ); and, by construction, BG is equal to GA; hence the triangles BGF, AGE have two angles and the included side of the one, equal to two angles and the included side of the other; they are, therefore, equal (Prop. For, if possible, let CD and CE be two perpendiculars; then, because CD is perpendicular to AB, the angle DCA is a right angle; _A B and, because CE is perpendicular to AB, C the angle ECA is also a right angle. Every page of this book bears marks of careful preparation. For the sake of brevity, the word line is often used to des Ignt'e a straight line. Moreover, the sum of the angles of the one polygon is equal to the sum of the angles of the other (Prop. Secondly, since ACB is an isosceles triangle, and the line CD bisects the base at right angles, it bisects also the vertical angle ACB (Prop. These two propositions, which, properly speaking, form but one, together with Prop. In an isosceles spherical triangle, the angles opposite the equal sides are equal; and, conversely, if two angles of a spherical triangle are equal, the triangle is isosceles. Now, because AE, CG are each of them parallel to BF, they are par- o allel to each other; therefore the diagonals AC, EG are in the same plane with AE, CG; and the plane AEGC divides the solid \ AG into two equivalent prisms.

If two solid angles are contained by three plane angles which are equal, each to each, the planes of the equal angles will be equally inclined to each other. Therefolre a circle may be described, &c. Scholium 1. Hence the two equal chords AB, DE are equally distant from the center. Draw AB perpendicular to DE; draw, also, the oblique lines AC, AD, AE. Therefore the three straight lines DE, DF, DG are equal to each other; and if a circumference be described from the center D, with a radius equal to DE, it will pass through the extremities of the lines DF, DG. Whence AVG is two thirds of ABVG; and the segment AVD is two thirds of the rectangle ABCD. For the same reason, BA and AH are in the same straight line. DF is equal to DIFF, and CD is equal to CDt; that is, the point D' is in the circumference of the circle ADA'G. Suppose, fol example, that the angles ACB, DEF are to each other as 7 to 4; or, which is the same thing, suppose that the angle M, which may serve as a common measure, is contained seven times in the angle ACB, and four times in the angle DEF. The arrangemleent of the propositions in this treatise is genlerally the same as in Legendre's Geometry, bult the form of the demonstrastions is reduced more nearly to the meodel of Euclid. Thus, if A:B: C:D; then, inversely, B: A. : D: C. Alternation is when antecedent is compared with antecedent, and consequent with consequent.

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