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ROBINSON'S SURVEYING AND NAVIGATION.

A Treatise on Surveying and Navigation, uniting the theoretical, the practical, and the educational features of these subjects. This work, in comparison with former works on the same subject, is greatly modernized and simplified. It contains also many collateral subjects. Its style and manner is such as to force itself upon the mind of the learner.

ROBINSON'S ELEMENTARY ASTRONOMY.

An abridged edition of the above, 228 pages, in which practical astronomy is not included, designed for a classbook in schools.

ROBINSON'S ASTRONOMY, UNIVERSITY EDITION.

In this work, facts are not only stated, but the manner of arriving at these facts is fully developed.

The subject of solar eclipses, both general and local, is more simple and comprehensively treated than in any former work; and simplicity, conciseness, and mathematical philosophy, are its distinctive characteristics.

Immediately after its publication it was adopted in the Normal Schools of Massachusetts, the Albany Academy, and many other leading institutions in the East.

ROBINSON'S CONCISE MATHEMATICAL OPERATIONS,

Being a sequel to the Author's Class Books, with much additional matter. A work essentially practical, designed to give the learner a proper appreciation of the utility of mathematics: embracing the gems of Science, from common Arithmetic through Algebra, Geometry, the Calculus, and Astronomy.

ROBINSON'S NATURAL PHILOSOPHY,

228 pages, in which there is more real philosophy than can be found in the same number of pages in any other book. Every principle is brought to the mind in a clear and practical point of view. This volume contains many philosophical problems to exercise the learner, and gives him a definite understanding of the principles of the steam engine, and is the only book which contains a full representation of the magnetic telegraph.

OF

GEOMETRY,

PLANE AND SPHERICAL TRIGONOMETRY,

AND

CONIC SECTIONS.

BY

H. N. ROBINSON, A. M.,

AUTHOR OF A TREATISE ON ARITHMETIC, AN ELEMENTARY AND A UNIVERSITY
EDITION OF ALGEBRA, A WORK ON NATURAL PHILOSOPHY, AND TWO SEPARATE
WORKS ON ASTRONOMY.

SIXTH STANDARD EDITION.

CINCINNATI:

JACOB ERNST, 112 MAIN STREET.

1854.

Entered according to act of Congress in the year 1850, by

H. N. ROBINSON,

In the Clerk's Office of the District Court of the United States, for the District of Ohio.

m

STEREOTYPED BY A. C. JAMES,

CINCINNATI.

PREFACE.

An attempt is made in this volume, to bring the science of geometry, directly to the comprehension of the learner; and to accomplish this end, it is necessary to sweep away some of the rubbish and some of the redundancies which have seemed only to obstruct our progress and becloud our vision.

All attempts to prove what is perfectly obvious to every one without proof, only weakens the mind rather than strengthens it, and hence, we have discarded all such propositions as the following: "All right angles are equal." Any two sides of a triangle are greater than the third side." "Parallel lines can never meet, however far they may be produced "-and some few others of like character. In almost every treatise on Geometry, the first, or one of the first propositions for demonstration is, "That all right angles are equal." This proposition at once excites in the mind of the intelligent pupil, a mingled sensation of disappointment and indignation,-disappointment, because he expected to learn new truths; indignation, because he feels as if his time and common sense are trifled with.

When he attempts the demonstration, he either has, or has not, a correct idea of a right angle; if he has a correct idea, he cannot demonstrate, or say anything that can be called a demonstration-because the proposition is all embraced in the definition of a right angle.

If he has not the correct idea of the term right angle, he must obtain it before he can commence any demonstration; so, in either case, the proposition is worse than useless.

When he comes to the proposition, that "Any two sides of a triangle, are together, greater than a third side,” and is carried through a useless demonstration, he looks about in wonder and perplexity, to discover why it is that he should be dragged through formal techicalities to arrive at the perfectly axiomatic truth, that a straight line is the shortest distance between two points.

Where is the logic of proving that parallel lines will never meet, however far they may be produced, when the very meaning of the term parallel is, that they cannot meet; hence, we say that all attempts to prove what is perfectly obvious, tend more to confuse and weaken, than to strengthen and enlighten.

Notwithstanding we have discarded such like propositions, we have omitted none of the truths therein expressed; for we have put them either in the axioms or definitions, and have made as complete a chain of geometrical truths as are to be found in any other work.

At the same time, no attempt has been made to present all the known propositions in geometry; we have taken such only as, united and combined, will give the pupil complete power over the science, and make his geometrical knowledge efficient, useful, and practical.

In the mathematical sciences, it is necessary to be more or less technical, formal, and exact; but we have made efforts not to be unpleasantly so. We have presumed that the reader will exercise his own judgment in construing our language; and in place of the preciseness of the professor, we have aimed to take the more wholesome and elevated tone of the practical common-sense man of the world. For the sake of perspicuity and brevity, we have freely used the algebraic language; and the whole work supposes that the reader clearly comprehends simple equations, and is able to perform all ordinary operations with them; but this should be no objection to the use of this book-for no treatise on Geometry should be studied prior to Algebra, whatever be the tone and style of the Geometry.

To most persons, Geometry is a very dry and uninteresting study; and from the nature of the human mind it must be so, until the pupil catches the spirit of the science; but as a general thing that spirit cannot be infused until some essential advancements have been made; hence, the ill success of many who undertake this study.

It is essential that the teacher should have a clear view of all these particulars; that he should possess the true spirit himself; and then he will be able to animate, encourage, and assist the new beginner, until the daylight of the science breaks in upon his mind.

It is of little use to commence Geometry unless the learner is determined to go through, at least, so far, as to understand Plane Trigonometry. The first propositions are only so many letters in the great alphabet of science, and we must be able to put them together, before we can really perceive their utility and power. These considerations induced us to be very full and practical in the application of Geometry, and if a student can go through this book understandingly, we are sure that his geometrical knowledge will be at once ample and efficient.