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THE

L O G. I. C. A N D U TIL IT Y

OF

MATHEMATICS.

WITH THE BEST METHODS OF INSTRUCTION EXPLAIN ED
AND ILLUSTRATED.

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NEW YORK :
PUBLISHED BY A. S. B A R N E S & CO.,
NO. 51 JOHN-STREET.
CINCINNATI:—H. W. DERBY & COMPANY.

Entered according to Act of Congress, in the year Eighteen Hundred and fifty, By CHARLES DAVIES,

In the Clerk’s Office of the District Court of the United States for the Southern District of New York.

STEREOTYPEID BY
RICHARD C. WALENTINE,

INEW YORK. **~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~...~~

F. C. GUTIERREZ, Printer,
No. 51 John-street, corner of Dutch.

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THE following work is not a series of speculations. It is but an analysis of that system of mathematical instruction which has been steadily pursued at the Military Academy over a quarter of a century, and which has given to that institution its celebrity as a school of mathematical science.

It is of the essence of that system that a principle be taught before it is applied to practice; that general principles and general laws be taught, for their contemplation is far more improving to the mind than the examination of isolated propositions; and that when such principles and such laws are fully comprehended, their applications be then taught as consequences or practical results.

This view of education led, at an early day, to the union of the French and English systems of mathematics. By this union the exact and beautiful methods of generalization, which distinguish the French school, were blended with the practical methods of the English system.

The fruits of this new system of instruction have been abundant. The graduates of the Military Academy have been sought for wherever science of the highest grade has been needed. Russia has sought them to construct her railroads;* the Coast Survey needed their aid; the works of internal improvement of the first class in our country, have mostly been conducted under their direction; and the recent war with Mexico afforded ample opportunity for showing the thousand ways in which science—the highest class of knowledge—may be made available in practice.

All these results are due to the system of instruction. In that system Mathematics is the basis—Science precedes Art— Theory goes before Practice—the general formula embraces all

the particulars.

It was deemed necessary to the full development of the plan of the work, to give a general view of the subject of Logic. The materials of Book I. have been drawn, mainly, from the works of Archbishop Whately and Mr. Mill. Although the general outline of the subject has but little resemblance to the work of either author, yet very much has been taken from both ; and in all cases where it could be done consistently with my own plan, I have adopted their exact language. This remark is particularly applicable to Chapter III., Book I., which is taken, with few alterations, from Whately.

For a full account of the objects and plan of the work, the reader is referred to the Introduction.

FISHKILL LANDING, {
June, 1850.

* Major Whistler, the engineer, to whom was intrusted the great enterprise of constructing a railroad from St. Petersburg to Moscow, and Major Brown, who succeeded him at his death, were both graduates of the Military Academy.

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LOGIC.

CHAPTER I.

DEFINITIONS—OPERATIONS OF THE MIND–TERMS DEFINED. . 27

Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Operations of the Mind concerned in Reasoning . . . . . .

Abstraction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Terms—Singular Terms—Common Terms. . . . . . . . . .

Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

SECTION

1— 6

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