Of Ratios and Proportions. THEOREM X. If four quantities are proportional, and one antecedent and its consequent be augmented by quantities which have the same ratio as the antecedent and consequent, the four quantities will still be in proportion. Let us take the proportions A : B :: C: D, and A: B :: E : F, which give in which the antecedem C and its consequent D, are augmented by the quantities E and F, which have the same ratio. Sch. The proposition may be verified by the proportion, 9 : 18 :: 20 : 40, in which the ratio is 2. If we augment the antecedent and its consequent by 15 and 30, which have the same ratio, we have If four quantities are proportional, and one antecedent and its consequent be diminished by quantities which have the same ratio as the antecedent and consequent, the four quantities will still be in proportion AxD-BXC and AXF-BX E. By subtracting these equalities, we have A×(D—F)=B×(C—E); and by Th. II, we obtain A : B :: C-E D-F, in which the antecedent and consequent, C and D, are ished by E and F, which have the same ratio. dimin Sch. The proposition may be verified by the proportion, for, by diminishing the antecedent and consequent by 15 and 30, we have If we have several sets of proportions, having the same ratio, any antecedent will be to its consequent, as the sum of the antecedents to the sum of the consequents. If we have the several proportions, A : B :: C : which gives AXD=BXC which gives AXF=BXE which gives AX H=BxG We shall then have, by addition, AX(D+F+H)=Bx(C+E+G); und consequently, by Th II. A B :: C+E+G : D+F+H. Of Ratios and Proportions. Sch. The proposition may be verified by the following proportions: viz. If four quantities are in proportion, their squares or cubes will also be proportional. If we have the proportion A: B :: C: D, it gives B D A C Then, if we square both members, we have B2 D2 = and if we cube both members, we have B3 D3 = and then, changing these equalities into a proportion, we have Sch. We may verify the proposition by the proportion, 2 : 4 :: 6 : 12, and by squaring each term we have, 4 : 16 :. 36 : 144 Of Ratios and Proportions. numbers which are still proportional, and in which the ratio If we have two sets of proportional quantities, the products o the corresponding terms will be proportional. and this by Th. II, gives AXE : BXF :: CX G : DXH. Sch. The proposition may be verified by the following proportions: 1 similar figures, are those which have the angles of the one qual to the angles of the other, each to each, and the aides about the equal angles proportional. 2. Any two sides, or any two angles, which are like placed in the two similar figures, are called homologous sides or angles. 3. A polygon which has all its angles equal, each to each, and all its sides equal, each to each, is called a regular polygon. A regular polygon is both equiangular and equilateral. 4. If the length of a line be computed in feet, one foot is the unit of the line, and is called the linear unit. If the length of a line be computed in yards, one yard is the linear unit 5. If we describe a square on the unit of length, such square is called the unit of surface. Thus, if the linear unit is one foot, one square foot will be the unit of surface, or superficial unit. 1 foot. unit |