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ALGEBRAICAL AND ARITHMETICAL ILLUSTRATIONS, EXPLANATORY NOTES,
AND A SYNOPTICAL INDEX TO THE SIX BOOKS, SHEWING

THE USES OF THE PROPOSITIONS, &c.,

BY HENRY GREEN, A. M.

“THERE IS (GENTLE READER) NOTHING (THE WORD OF GOD ONELY SET APART) WHICH SO
MUCH BEAUTEFIETH AND ADORNETH THE SOULE AND MINDE OF MAN, AS DOTH THE KNOWLEDGE
OF GOOD ARTES AND SCIENCES.”

Billingsley's Euclid, A. D. 1570.

MANCHESTER: JOHN HEY WOOD, 143, DE ANSGATE.

LONDON : SIMPKIN, MARSHALL, & CO.

1861.

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PREFACE..

The Preface to Bks. I and II of the “Gradations in Euclid" contains most of the observations which can be required from the Author. One of them he now repeats ;—“At the present day nearly every edition of Euclid's Elements must be more or less a compilation, in which the Author draws freely on the labours of his predecessors. “The Gradations' are, in a great degree, of this character; and an open acknowledgment will suffice, once for all, to repel any charge of intentionally claiming what belongs to others. It is affectation to pretend to great originality on a subject which has, like Geometry, for so many centuries exercised men's minds.”

Originality has not been the Author's aim, but usefulness. With this purpose before him, he has endeavoured to show how few, if any, Geometrical Truths are destitute of a practical application. The stigma which some have attempted to fasten on what they name, "mere Mathematical Theories,” is thus removed, and the Science, instead of being repulsive from its dry abstractions, is invested with the ever-abiding charm of being both the foundation and the builder-up of very many most important practical results. In fact, there is scarcely a branch of human knowledge, from the art of sketching an outline, to that of spanning and measuring the heavens themselves, which does not depend for its vigour and comiprehensiveness on the aids which Geometry furnishes.

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