...takes in travelling the whole journey. 7. Prove that when any number of quantities are proportionals, as one antecedent is to its consequent, so is the sum of all the antecedents to the sum of all the consequents. 8. Extract the square root of — 47-12\/l5....

...of magnitudes of the same kind are proportionals, as one of the antecedents is to its con_ sequent, so is the sum of the antecedents to the sum of the consequents. Let A,C,E . . . and B,D,F ... be magnitudes of the same kind which are proportionals, so that A:B::C:D::E:F:...

...the A LFG :: the A EBC : the A LGH :: the A ECD : the A LHK. But in a series of equal ratios, as each antecedent is to its consequent so is the sum of the antecedents to the sum of the consequents; [Addenda. V. 12.] .-. the A EAB : the A LFG : : the fig. ABCDE : the fig. FGHKL. (iii) Now the A EAB...

...is to FD, therefore, alternately, AE is to CFa& EB is to FD. [v. 16] And, as one of the antecedents is to its consequent, so is the sum of the antecedents to the sum of the consequents : [v. 12] therefore, as EB is to FD, so are AE, EB together to CF, FD together ; that is, AB is to...

...A>C and therefore B>D, but to the ratio С - A : D - В when A< С and therefore B< D. Or, in umrds, as one antecedent is to its consequent, so is the...is obvious, since in the same notation as before. A + G and therefore This theorem is a particular case of a more general theorem, which may be stated...

...CF'v& to FD, therefore, alternately, AE is to CFas EB is to FD. [v. 16] And, as one of the antecedents is to its consequent, so is the sum of the antecedents to the sum of the consequents : [v. 1 2] therefore, as EB is to FD, so are AE, EB together to CF, FD together; that is, AB is to...

...proportion is often used in Dynamics. Article VIII. — When any number of quantities' are proportional, as one antecedent is to its consequent, so is the sum of all the antecedents to the sum of ail the consequents ; that is, if a : b : : c : d : : e : f &c.,...