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THEORY AND PRACTICE.
THE SECOND EDITION, WITH MANY ADDITIONS.
By CHARLES HUTTON, LL. D. F. R. S. &c. &c.
PROFESSOR OF MATHEMATICS IN THE ROYAL MILITARY
PRINTED FOR G. G. J. AND J. ROBINSON, AND R. BALDWIN,
EARL AND BARON PERCY, LUCY, POYNINGS, FITZPAYNE, BRYAN, LATIMER, AND WARKWORTH, AND BARONET;
LORD LIEUTENANT AND CUSTOS ROTULORUM OF THE COUNTY OF NORTHUMBERLAND, AND OF THE TOWN AND COUNTY OF THE TOWN OF NEWCASTLE UPON TYNE; VICE ADMIRAL
OF THE COUNTY OF NORTHUMBERLAND; AND
LIEUTENANT GENERAL OF HIS MAJES
TY'S FORCES, COLONEL OF THE
SECOND TROOP OF HORSE
&c. &c. &c.
F the high honour this Work formerly received in the countenance of Your Grace's illuftrious Father, had not particularly encouraged the Author's prefumption, in thus feeking the protection of his noble Representative, he is convinced that he could not easily have found among the great and elevated, a character under whofe aufpices
aufpices he should more earneftly have wished to prefent it to the world in this its improved state.
The great progrefs of all the useful arts and fciences in this country, fince the happy æra of the acceffion of our auguft Sovereign, must be afcribed, next to his benign and munificent influence, to that eminent countenance which fome of the greatest characters have fhewn, by their perfonal example in the cultivation of them, not lefs than by their patronage and protection.
To the hereditary fame of Your Most Noble Progenitors, Your Grace's perfonal bravery and military talents have added great and diftinguished luftre. The public confidence and efteem which neceffarily follow fuch accomplishments, ftamp a value on every thing, however inconfiderable, which is honoured with Your Grace's patronage: And the accurate judgment which Your Grace is known to poffefs on all fubjects of extenfive practical utility, makes it the wifh of fuch as cultivate them, ardently to feek Your Grace's protection.
With these impreffions, I prefume to lay the following performance at Your Grace's feet; and am, with the profoundeft refpect,
moft obedient, and
moft devoted humble fervant,
BY Menfuration I understand the art and science which is con
cerned about the measure of extenfion, or the magnitude of figures; and it is, next to arithmetic, a subject of the greateft ufe and importance, both in affairs that are abfolutely neceffary in human life, and in every branch of the mathematics: a fubject by which sciences are established, and commerce is conducted; by whofe aid we manage our business, and inform ourfelves of the wonderful operations of nature; by which we measure the heavens and the earth, estimate the capacities of all veffels, and bulks of all bodies: gauge our liquors, build edifices, measure our lands and the works of artificers, buy and fell an infinite variety of things neceffary in life, and are fupplied with the means of making the calculations which are necellary for the conftruction of almost all machines.
It is evident that the clofe connection of this fubject with the ordinary affairs of life, would very early evince its importance to mankind; and accordingly we find, that the most celebrated philofophers have paid the greatest attention to it; and the chief and most effential discoveries in geometry in all ages, have been made in confequence of their attempts to improve this fcience. 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 feem to have had no other end befides menfuration in view, in all their laboured geometrical difquifitions. 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 feems to have adapted them entirely to this purpose. For, if it be confidered that his Elements contain a continued chain of reasoning, and of truths, of which the former are fucceffively applied to the discovery of the latter, one propofition depending on another, and the fucceeding propofitions ftill 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 fcarcely