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its ſuperficies; a thing which is in my judgment well werth obſerving, and perform'd, as I know of, by none.
Firſt therefore, for finding the ſolidity of a ſegment, I ſhall lay down two, commonly known and receiv'd Suppoſitions, viz.
1. That a ſeries of magnitudes proceeding in Arithmetical Progreſſion from nothing (incluſive) or whoſe common difference is equal to the leaſt magliitude, is ſubduple of as many quantities equal to the greateſt; (i. e. ſubdupie of the product of the greateſt term and number of terms:) So that if the ſum of the terms be called z, the greateſt term g, and the number of terms n, then will z = ng2 , or 2 z = ng.
The truth of this Propoſition will eaſily appear by expreſſing the ſeries twice, and inverting the order;
0, a, 2a, 3a, 4a.
4a, 3a, 2a, a, 0.
For ſo the difference always being equal to the leaſt quantity, 'twill be evident that each two correſpondent terms taken together are equal to the greateſt term; and alſo, that the ſeries taken twice is equal to the greateſt term repeated as many times as there are terms, i. e. the laſt term drawn into the number of terms.
We have in a triangle a very clear and eaſy example of this moſt uſeful Propoſition, which is prov'd hence, to he half a parallelogram having the ſame altitude, and ſtanding on the ſame baſe.
