Page:Euclid's Elements 1714 Barrow translation.djvu/515

 its uperficies; a thing which is in my judgment well werth oberving, and perform'd, as I know of, by none.

Firt therefore, for finding the olidity of a egment, I hall lay down two, commonly known and receiv'd Suppoitions, viz.

1. That a eries of magnitudes proceeding in Arithmetical Progreion from nothing (incluive) or whoe common difference is equal to the leat magliitude, is ubduple of as many quantities equal to the greatet; (i. e. ubdupie of the product of the greatet term and number of terms:) So that if the um of the terms be called z, the greatet term g, and the number of terms n, then will z = $ng⁄2$, or 2 z = ng.

The truth of this Propoition will eaily appear by expreing the eries twice, and inverting the order;

For o the difference always being equal to the leat quantity, 'twill be evident that each two correpondent terms taken together are equal to the greatet term; and alo, that the eries taken twice is equal to the greatet term repeated as many times as there are terms, i. e. the lat term drawn into the number of terms.

We have in a triangle a very clear and eay example of this mot ueful Propoition, which is prov'd hence, to he half a parallelogram having the ame altitude, and tanding on the ame bae.