convertendo, AC :·(AD—AC) :: i. e AC CD:: AG: GB. : AG: (AB-AG); EXAMPLE 1. Divide the line AB, whose length is 48, into the following proportional parts, 6 to 2; 7 to 1; and 5 to 3. ANSWER. 36, 12; 42, 6; 30, 18. USEFUL THEOREMS. 1. If two straight lines cut one another, the opposite angles are equal, and the four angles are together equal to four right angles; i. e. the angle AEC is equal to the angle DEB, and A E the angle CEB to the angle AED. (Euclid I, 15.) 3. All the interior angles of any rectilineal figure are equal to four less than twice as many right angles as the figure has sides; i. e. in the five-sided figure ABCDE, all the interior angles at A,B,C,D, and E, are A E B E equal to four less than twice five right angles; that is, are equal to six right angles or 540 degrees. (Euclid I, 32, Cor. 4. The greatest angle of every triangle is opposite the greatest side. (Euclid I, 18.) 5. Parallelograms and triangles upon equal bases and between the same parallels, that is, having the same perpendicular height, are equal to each other; i.e., the pa- B D E C rallelogram ABCD=the parallelogram DBCE; and the triangle ABC=the triangle CFE. (Euclid I, 35-38.) 6. In every right-angled triangle, the square of the hypothenuse is equal to the sum of the squares of the base and perpendicular; i.e., in the right-angled triangle ABC, AC AB2+BC. B A D segments of the hypothenuse; and the base and perpendicular are, respectively, mean proportionals between the hypothenuse and the segment adjacent to them; ie., BD. DC=DA2; CB. BD-BA2; and BC. CD=CA3. (Euclid VI, 8.) Also BAAC2 BD - DC2 or BA+AC. BA—AC=BD+DC. BD-DC. (Euclid II. 5 Cor.) This is a very useful proposition in measuring the width C of a river; one or two practical applications of it will be given in a subsequent part of the work. 9. The angle, in a semi-circle, is a right angle; in a segment, less than a semi-circle, greater than a right angle; in a segment, that is greater, less; ie., AB being the diameter, the angle AEB is a right angle; the angle EFB, being in a segment less A B F D than a semi-circle, is greater than a right angle; and the angle BAE is less. (Euclid III, 31.) 10. Angles, standing upon equal circumferences, are equal, whether they be at the centre or the circumference; i.e., if the circumference BE be equal to the circumference BC, the angles EAB, BAC on the circumference, and the angles E B FDB, BDC at the centre, are equal; and the angles EDBBDC are respectively double the angles EAB, BAC. (Euclid III, 20 and 27.) 11. If a line touch a circle, any angle made between this line at the point of contact, and a line cutting the circle, is equal to the angle in the alternate segment; i.e. (See Fig. 9), the angle CBE equals the angle EFB, and the angle CBA equals the angle AEB. (Euclid III, 32.) 12. Parallelograms, and triangles of equal altitude, are as their bases; and of equal bases, are as their altitudes. (Euclid VI, 1.) And, when neither are equal, in the com, pound ratio of both; i.e., if there be two triangles whose bases and perpendiculars are as follows: the one base 40, the other 20; the one height 30, and the other 15; the area of the larger triangle is to that of the smaller, as 40×30 is to 20×15, or as 1200 to 300; or as 4 to 1. 13. Similar triangles have their areas, proportional to the squares of their homologous sides. (Euclid VI, 19.) That is, the area of the triangle EBF is to that of the triangle ABC, as EB2 is to AB2, or as BF2 is to BC2. PRELIMINARY OBSERVATIONS. The statute acre in England consists of ten square chains, that is, of ten square blocks of land, whose length and breadth are one chain each; or, taking the 10 blocks as one whole block, then an acre is equal to the area of a rectangle, whose base is 10 chains and perpendicular 1 chain; or to any other rectangle, the product of whose base and perpendicular also equals 10 chains: thus, 4 chains base with 2 chains perpendicular, and 2 chains base with 5 chains perpendicular are each equal to 1 acre. |