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By CHARLES HUTTON, LL. D. F. R. S. &c. &c.
PRINTED FOR G. G. J. AND J. ROBINSON, AND R. BALDWIN,
ST. PAUL'S CHURCH-YARD.
T 0 THE MOST NOBLE AND PUISSANT PRINCE,
H U G Ꮐ . H,
DUKE AND EARL OF NORTHUMBERLAND,
LARL AND BARON PERCY, LUCY, POYNINGS, FITZPAYNE, BRYAN,
LATIMER, AND WARKWORTH, AND BARONET;
LORD LIEUTENANT AND CUSTOS ROTULORUM OF THE COUNTY OF
&c. &c. &c.
F the high honour this Work formerly re
ceived in the countenance of Your Grace's illustrious Father, had not particularly encouraged the Author's presumption, in thus seeking the protection of his noble Representative, he is convinced that he could not easily have found among the great and elevated, a character under whose
auspices he should more earnestly have wished to present it to the world in this its improved state.
The great progress of all the useful arts and fciences in this country, since the happy æra of the accession of our august Sovereign, must be ascribed, next to his benign and munificent influence, to that eminent countenance which some of the greatest characters have shewn, by their fonal example in the cultivation of them, not less than by their patronage and protection.
To the hereditary fame of Your Most Noble Progenitors, Your Grace's personal bravery and military talents have added great and distinguished lustre. The public confidence and esteem which necessarily follow such accomplishments, stamp a value on every thing, however inconsiderable, which is honoured with Your Grace's patronage : And the accurate judgment which Your Grace is known to possess on all subjects of extensive practical utility, makes it the wish of such as cultivate them, ardently to seek Your Grace's protection.
With these impressions, I presume to lay the following performance at Your Grace's feet; and am, with the profoundest respect,
most obedient, and
cerned about the measure of extension, or the magnitude of figures ; and it is, next to arithmetic, a subject of the greateft use and importance, both in affairs that are absolutely necessary in human life, and in every branch of the mathematics: a subject by which sciences are established, and commerce is conducted; by whose aid we manage our business, and inform ourselves of the wonderful operations of nature; by which we measure the heavens and the earth, estimate the capacities of all vessels, and bulks of all bodies: gauge our liquors, build edifices, measure our lands and the works of artificers, buy and sell an infinite variety of things necessary in life, and are supplied with the means of making the calculations which are necellary for the construction of almost all machines.
It is evident that the close connection of this subject with the ordinary affairs of life, would very early evince its importance to mankind ; and accordingly we find, that the most celebrated philosophers have paid the greatest attention to it; and the chief and moit essential discoveries in geometry in all ages, have been made in consequence of their attempts to improve this science. Socrates thought that the prime use of geometry was to measure the ground, and indeed this business gave name to the subject; and most of the ancients seem to have had no other end besides mensuration in view, in all their laboured geometrical disquisitions. Euclid's Elements are almost entirely devoted to it; and although there be contained in them many properties of geometrical figures which may be applied to other purposes, and indeed of which the moderns have made the greatest use in most other sciences; yet Euclid himself seems to have adapted them entirely to this purpose. For, if it be considered that his Elements contain a continued chain of reasoning, and of truths, of which the former are fucceflively applied to the discovery of the latter, one proposition depending on another, and the fucceeding propofitions still approximating towards fome particular object near the end of each book; and when at the last we find that object to be the equality, proportion, or relation between the magnitudes of figures, both plane and folid; it is scarcely