...terms of a Progression, are called the Extremes ; and the other terms, lying between them, the Means. The most useful part of arithmetical proportions,...quantities are in arithmetical proportion, the sum uf the two extremes is equal to the sum of the two means. Thus, of the four 2, 4, 6, 8, here 2 48 =...

...Progression the numbers or terms have all the same multiplier or divisor. The most useful part of Proportion, is contained in the following theorems. THEOREM 1. When four quantities are in proportion, the product of the two extremes is equal to the product of the two mean?. Thus, in the...

...proportions. THEOREM 3. In an arithmetical series, consisting of 4, 6, or any even number of terms, the sum of the extremes is equal to the sum of the two cciddle terms, or to the sum of any two means equally distant from the extremes. Thus, in the series...

...plus 4. The reasoning would be the same for every other arithmetical proportion. Therefore, in every arithmetical proportion, the sum of the extremes is equal to the sum of the means. If the arithmetical proportion were continued, it is evident that the sum of the extremes would...

...terms of a Progression, are called the Extremes ; and the other terms, lying between them, the Means. The most useful part of arithmetical proportions, is contained in the following theorems : THEOREM I. When four quantities are in arithmetical proportion, the sum of the two extremes is equal to the...

...deduced from the nature of the series, in the following manner. As we found in arithmetic proportion that the sum of the extremes is equal to the sum of the means, so it is evident that here the sum of the extremes is equal to the sum of any two terms equally...

...transposing— b and —m, a+m=6+A So in the proportion, 12..10: ;11..9, we have 12+9 = 10+11. Again, if three quantities are in arithmetical proportion, the sum of the extremes is equal to double the mean. If a . . 6: '.b .. c, then, a — b=b — c S And transposing - 6 and — c, ' o+c—...

...with the equation 6 — o = d — c, from which we deduce a -f- d = 6 -f- c Thus in an equidifference the sum of the extremes is equal to the sum of the meant. This is the leading property of equi•differences. Reciprocally, let there be four quantities...

...and the other terms, lying between them, the Means. The moat useful part of arithmetical proportion, is contained in the following theorems : THEOREM 1....quantities are in arithmetical proportion, the sum of the two extremes is equal to the sum of the two means. Thus, of the four 2, 4, 6, 8, here 2 48 = 4 + 6=10....

...separate consideration. The proportion a..b::c..d It will be proper, however, to observe that, if fowr quantities are in arithmetical proportion, the sum of the extremes is equal to the sum of the means. Thus if a . . b : : h . . in, then ,a-\-m= b-\-h For by supposition, a - b = h - m And transposing...