Εικόνες σελίδας
PDF
Ηλεκτρ. έκδοση

In this work (Fig. 22 C) the triangle is placed in its natural position (an important matter to a young student), and direct proof is given with the entire quantities; a rectangle forming part of the square on the hypothenuse and the square on one side being proved (by Sect. 17) to be respectively equivalent with a given parallelogram (A), and therefore equivalent between themselves (Axiom 1); and then the rectangle completing the square on the hypothenuse, and the square on the third side, are proved (also by Sect. 17) to be respectively equivalent with another parallelogram (B), and therefore equivalent between themselves; and by a simple addition (Axiom 2) the square on the hypothenuse is shown to be obviously eqivalent with the squares on the other sides.

Not only is the demonstration I give more clear and direct than that in Euclid, but it naturally suggests a deduction of the very greatest importance in geometrical operations, though not given in Euclid-namely, that "The squares on the two sides bear the same proportion to the adjoining segments of the hypothenuse as the square on the hypothenuse bears to the whole line."

This deduction, with the proposition that "The areas of similar figures are proportional to the squares of their corresponding sides or diagonals" (given in Div. F), form the keystone to the most interesting and useful doctrine of proportionals, as regards the areas of surfaces.

Sect. 27 is also applied to many and important matters to which it is not applied in Euclid, including the doctrine of "Obliques" and the comparative lengths of "Chords in Circles;" attaining results by a single tool where many are used in Euclid.

The remaining Sections of Div. C (28 to 45) make it complete in any questions likely to arise as regards "Equivalents."

In the subsequent Divisions, the operations are also chiefly grounded on deductions from the original base.

DIVISION D. Differentials. This Division, as its name implies, has relation to the differences between figures; it comprises a number of interesting and useful propositions, chiefly deduced from French works, except Sectns. 14 to 22, which comprise propositions 2 to 10 in the 2nd Book of Euclid, which are most useful, but as given in Euclid are difficult to attain and still more difficult to retain.

I have endeavoured to make them clear and interesting to a student.

DIVISION E. Measurement of Angles. This Division treats the subject in an entirely new manner, and substitutes for the several, and at first view apparently contradictory rules, in Euclid, a simple and clear rule under which angles generally may be measured (see p. 143, &c., 2 E). Instructions for measurement of "Reflex" angles, or those exceeding half a circle, are also given.

Any angle may be measured as within a circle by taking the summit as centre, and with any radius drawing an arc to cut the sides, and Sect. 11 of the Division will then apply.

DIVISION F. Proportionals. In this most interesting and useful branch of geometry, extending to 55 Sections and upwards of 40 figures, great care has been taken to treat the subject, both scientifically and practically, as fully and clearly as its very great importance deserves, and also to introduce many new and suggestive propositions and figures.

DIVISION G. Partitionals. The matter contained in this Division is almost ignored in Euclid, and is seldom attended to in English works, though very carefully given in several French works, and especially that by Colonel Guy.

The subject is practically of great importance in France, from the minute subdivision of land, and not only calls for the division into several parts equal in quantity, but often for the effecting the division so as to allow every part to join a pond, well, road or river, or other appendage with which every part must have a junction.

As geometrical exercises, partitions are most interesting, and the operations almost entirely depending on Sect. 17, Div. C, even a young student may readily solve problems in this most interesting Division.

I give many interesting modes of partitions, but almost endless varieties might be introduced for pleasurable mental exercise.

Having taken very great pains in preparing the corresponding Division in my "Practical Geometry," I have in great measure followed that Division in the present work.

DIVISION H. Mensuration. For the like reason this Division is in great part taken from the corresponding Division in “Practical Geometry;" and although not strictly within the original scope of the present work, I have retained the parts applicable to the measurement of distances, altitudes, and angles, and also of solids, or cubic

measure.

I intended to have added some remarks on the earlier parts of Euclid's work, to show its unsuitableness for the

instruction of young students, but space will not allow me to do more than point out how unwise it is to compel a young student to solve the 5th proposition in the 1st Book, and in a manner most difficult to him (elegant as the solution is to one acquainted with geometry), whilst by simply teaching the mechanical operation of bisecting an angle, two equal triangles are shown, having obviously equal angles at the base, and also angles below the base equal as being supplementary.

Proposition 7 has also a very long, and to a student a very difficult solution; whilst by drawing semicircles as in Fig. 2 A, the equality of radii will obviously demonstrate the proposition.

The greater portions of this work are entirely new, including the introduction of the scientific base; the demonstration in Sect. 27 in Div. C and the numerous and interesting deductions from it, and in the arrangement of the Divisions, and the giving a mode of ascertaining directly the proportions in the sides of any triangle (page 177, &c.).

Throughout the work, I have endeavoured to make the demonstrations so clear as to be understood by anyone who has never before studied geometry.

As to my RANGE FINDER; it is based on a rhombus, having sides of fifty yards each, up to 4,000 yards, and of 100 yards up to more than six miles; and for distances up to 2,000 yards, sides of twenty-five yards would be sufficient. The operations are so simple, that less than an hour's study would give the requisite information.

Performed by three privates and a non-commissioned

officer, the four measurements in Fig. 8 H might be completed in three minutes; and for coast defence, the distance of a ship moving at full speed might be ascertained in less than a third of a minute.

The apparatus for distances to 4,000 yards is contained in a japanned case, 24 inches long, 4 inches broad, and 1 inch

deep.

The weight, including a shoulder-strap, is about four pounds, and the expense made in numbers would be less than 108., or as to the apparatus for the six miles range would be between 30s. and 21.

By the addition of two bulls-eye lanterns, the measurements to a light in the distance might be made as readily by night as by day.

I have spared no pains to make the work as attractive as I trust it will be useful, and sincerely hope that in the attempt I have been successful.

FERN HILL, MELTON, WOODBRIDGE,

June, 1879.

ROLLA ROUSE.

« ΠροηγούμενηΣυνέχεια »