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E U C L I D's
E L E M E N T S

OF

GEOMETRY,
The First Six, the Eleventh and Twelfth

Β Ο Ο Κ

Si

Translated into ENGLISH,

From Dr. GREGORY's EDITION, LITEE

WITH

NOTES and ADDITIONS,

For the Use of the BRITISH YOUTH,

By E. S T O N E.

The Second Edition, with ADDITIONS.

Ptolemy, King of Ægypt, having asked Euclid whether there

was any other more compendious way of arriving at Geometry,
than by these his Elements, is said to bave answered: M,
είναι βασιλικήν ατραπόν επί γεωμεθρίαν, There is no other way,
ar royal paffage, to Geometry.

Proclus's Commentary upon Euclid's second book,

L O. N D O N,
Printed for John RIVINGTON, in St. Paul's Church-Yard,

WILLIAM JOHNSTON, in Ludgate-Street, and THOMAS
LONGMAN, in Pater nofter-row.

M.DCC.LXY.

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Isdom is certainly one of the main springs

of human happiness. And the more of ic each individual of a society acquires, the more happy, all things else alike, will that society be ; for the wiser a man is, the better will he know how to preserve his own health, and pleasantly lengthen out his own life: he will better avoid the injuries of weather ; better provide food and cloathing; better avoid pain and sickness; betcer sooih, lessen, and command his own and other people's passions ; better observe the laws of his country; and, in short, better promote his own and neighbours happiness in all respects soever. Accordingly, since wildom is so necessary to happiness, that learning, which is best adapted to promote and extend it, is undoubtedly the most valuable, and ought by all to be most encouraged, cultivated, and used. And therefore, since true wisdom very much consists in, and is ob tained from, the knowledge of the comparative numbers and magnitudes of the qualities, powers, efficacies, matter, and motions of sensible objects, their various increase and decrease, &c. and as these may be acquired in a great measure from arithme, tic and geometry: these two sciences may truly be said to be the two great fountains from whence much of the wisdom and happiness of mankind flow: And therefore, if possible, they should be learned by every member of a society. No state ever did or scarcely could flourish without them. They have always been in esteem and cultivated proportionably to the greatness and riches of a country, and as these have declined, so have those,

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