THEOREM VI. Triangles standing on the same base, and between the same parallels are equal to each other. Let the triangles ABC, DBC be upon the same base BC, and between the same parallels AD, BC; the triangle ABC is equal to the triangle DBC. (Euc. I. 37. Simp. II. 2. Em. II. 10.) B THEOREM VII. In any right-angled triangle, the square of the hypothenuse is equal to the sum of the squares of the other two sides. Let ABC be a right-angled triangle, having the right angle BAC; the square of the side BC is equal to the sum of the squares of the sides AB, AC. (Euc. I. 47. Simp. II. 8. Em. II. 21.) A NOTE. Pythagoras, who was born about 2400 years ago, discovered this celebrated and useful Theorem; in consequence of which, it is said, he offered a hecatomb to the gods. THEOREM VIII. An angle at the centre of a circle is double the angle at the circumference, when both stand on the same arc. Let ABC be a circle, and BDC an angle at the centre, and BAC an angle at the circumference, which have the same arc BC for their base; the angle BDC is double of the angle BAC. (Euc. III. 20. Simp. III. 10. Em. IV. 12.) THEOREM IX. An angle in a semicircle is a right angle. Let ABC be a semicircle; then the angle ABC in that semicircle is A a right-angle. (Euc. III. 31. Simp. III. 13. Em. VI. 14.) B THEOREM X. If a line be drawn in a triangle parallel to one of its sides, it will cut the two other sides proportionally. In the preceding figure, DE being parallel to BC, the triangles ABC, ADE are equi-angular or similar; therefore, AB is to BC, as AD to DE; and AB is to AC, as AD to AE. (Euc. VI. 4. Simp. IV. 12. Em. II. 13.) THEOREM XII. In a right-angled triangle, a perpendicular from the right angle is a mean proportional between the segments of the hypothenuse; and each of the sides, about the right angle, is a mean proportional between the hypothenuse and the adjacent segments. A Let ABC be a right-angled triangle, having the right. angle BAC; and from the point A let AD be drawn perpendicularly to the base BC; the triangles ABD, ADC are similar to the whole triangle ABC, and to each other. Also the perpendicular AD is a mean proportional between the segments of the base; and each of the sides is a mean proportional between the base and its segment adjacent to that side; therefore, BD is to DA, as B D с DA to DC; BC is to BA, as BA to BD; and BC is to CA, as CA to CD. (Euc. VI. 8. Simp. IV. 19. Em. VI. 17.) THEOREM XIII. Similar triangles are to each other as the squares of their like sides. A Let ABC, ADE, be similar triangles, having the angle A common to both; then the triangle ABC is to the triangle ADE, as the square of BC to the square of DE. That is, similar triangles are to one another in the duplicate ratio of their homologous sides. (Euc. B VI. 19. Simp. IV. 24. Em. II. 18.) THEOREM XIV. D E C In any triangle the double of the square of a line drawn from the vertex to the middle of the base, together with double the square of half the base, is equal to the sum of the squares of the other sides. In any triangle ABC, double the square of a line CD, drawn from the vertex to the middle of the base AB, together with double the square of half the base AD or BD, is equal to the sum of the squares of the other sides AC, BC. (Simp. II. 11. Em. II. 28.) THEOREM XV. In any parallelogram ABCD, the sum of the squares of the two diagonals AC, BD, is equal to the sum of the squares of all the four sides of the parallelogram. (Simp. II. 12. Em. III. 9.) THEOREM XVI. A B All similar figures are in proportion to each other as the squares of their homologous sides. (Simp. IV. 26. Em. III. 20.) THEOREM XVII. The circumferences of circles, and the arcs and chords of similar segments, are in proportion to each other, as the radii or diameters of the circles. (Em. IV. 8 & 9.) THEOREM XVIII. Circles are to each other as the squares of their radii, diameters, or circumferences. (Em. IV. 35.) THEOREM XIX. Similar polygons described.in circles are to each other, as the circles in which they are inscribed; or as the squares of the diameters of those circles. (Em. IV. 36.) THEOREM XX. All similar solids are to each other, as the cubes of their like dimensions. (Em. VI. 24.) AN EXPLANATION OF THE . PRINCIPAL MATHEMATICAL CHARACTERS. = THE sign or character (called equality) denotes that the respective quantities, between which it is placed, are equal; as 4 poles 22 yards = 1 chain 100 links. = The sign (called plus, or more) signifies that the numbers, between which it is placed, are to be added together; as 9 + 6 (read 9 plus 6) = : 15. Geometrical lines are generally represented by capital letters; then ABCD signifies that the line AB is to be added to the line CD The sign (called, minus, or less) denotes that the quantity, which it precedes, is to be subtracted; as 15 – 6 (read 15 minus 6) 9. In geometrical lines also, AB– CD signifies that the line CD is to be subtracted from the line AB. = The sign denotes that the numbers, between which it is placed, are to be multiplied together; as 5 x 3 (read 5 multiplied by 3) = 15. The sign signifies division; as 153 (read 15 divided by 3) 5. Numbers placed like a vulgar fraction, also denote division; the upper number being the dividend, and the lower the divisor; as 15 3 = 5. The signs (called proportionals) denote proportionality; as 2:5:: 6: 15, signifying that the number 2 bears the same proportion to 5, as 6 does to 15: or, in other words, as 2 is to 5, so is 6 to 15. The sign (called vinculum) is used to connect several quantities together; as 9 + 3 6 x 2 = 6 x 2 = 12. – 6 | ×. The sign 2, placed above a quantity, represents the square of that quantity; as 5+32=828 x 8 = 64. The sign, placed above a quantity, denotes the cube of that quantity; as 93-83 = 12 −8 |3 = 43 = 4 × 4 x 4 = 64. The signor, placed before a quantity, denotes the square root of that quantity; as 9 x 4 = √36 = 6. The sign, placed before a quantity, represents the cube root of that quantity; as 36 × 4 x 3. 8= Angle; as A, signifies the angle A; or the angle ABC, B denotes the angular point. |