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the portion devoted to Solid Geometry, where the attempt is made to “educate students mainly through their own work.” They are encouraged to make their own models of solids and of figures in solid geometry in such a way as not only to verify their answers, but to bring before them clearly in the concrete what had before existed to them only in the abstract. Full directions as to the manner in which the models should be made are given in every case.

The advantages accruing from this method of learning the subject are so obvious, that it is hoped that what has been singularly successful in a small sphere only will now become more general and wide in its adoption.

To give further aid to the reader, photographs of actual models have been inserted just at those places where the average student finds most difficulty.

Another feature of the book consists of the numerous notes and explanations, printed in small type, and inserted immediately after the text to which they refer; it is desirable that these notes should be read, as they often contain hints and cautions which tend to furnish the memory with important facts. In short, the general scheme of the book will, it is confidently believed, induce teachers to avoid unnecessary class-lecturing, and encourage personal study on the part of the student. The three latest Examination papers in Elementary Practical Plane and Solid Geometry, set by the Board of Education, have been appended at the end of the book.

In conclusion, the author wishes to record his indebtedness to Mr. Joseph Knight, of the Technical School at Bury, for many suggestions in the preparation of the manuscript.

G. F. B,
LEEDS TECHNICAL School, 1903.

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PLANE AND SOLID GEOMETRY.

PART 1.-PLANE GEOMETRY.

CHAPTER I.

INTRODUCTION.-TERMS USED AND

DEFINITIONS. NEATNESS and accuracy of draughtsmanship are indispensable. Care should be taken by the student in the selection of his apparatus, as it is very easy for a beginner to go astray or be misled into purchasing worthless or unnecessary drawing instruments. On the whole, it is advisable that he should consult his teacher before making any purchase. The following apparatus is necessary for the Elementary Stage.

(1) A Drawing-Board. This should be not less in size than Half Imperial (23 inches by 16 inches), on which smooth cartridge drawing paper should be fixed at the corners with four drawing pins.

(2) A T-Square. The size should correspond to that of the drawing-board, and should be used with the butt-end in contact with the left side of the board. The blade ought not to be thick, and must possess a straight edge.

(3) Pencil Compasses. This instrument should have a firm needle-point without any shake whatever, and a slightly rounded shoulder to the needle holder. A convenient size, with knee-joints, is 3". Should the student desire to ink in his drawings with Indian ink, a half-set of drawing instruments together with a drawing pen are the most economical.

(4) Two Set-Squares. One has angles of 90°, 45°, and 45°, and the other 90°, 60°, and 30°. For sizes, the bases may be about 5" in length. They should be made of rather thin

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PR. GEO.

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vulcanite, and the angles quite accurate. Suppose paper is fixed on the drawing-board. To draw lines on the paper А

parallel to AB (see fig. 1) B

the T-square alone is used. In the case of lines perpendicular, as those marked 1, 2, 3, as well as inclined parallel lines, as 4, 5, 6, a setsquare is used, and it is guided either with the

edge of the T-square as Fig. 1.

shown, or with the edge of the other set-square, whichever may be most convenient at the time.

(5) Pencils. Generally he is the best ; one end should have a long chisel-point, and be always kept with a keen edge by means of fine sand-paper or a file. As solutions to problems are to be drawn in strong lines, it is advisable to use an Ab pencil for that purpose.

(6) A Protractor. The rectangular protractor (fig. 2) is very serviceable if it have {ths and uths of an inch, as well as a diagonal scale (see page 37), on the reverse face.

It is used when lines are to be drawn inclined to one another at given angles. Thus, suppose a line ac is required

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to be drawn from A (fig. 3) at 40 degrees (40°) to AB. Place the base of the protractor at AB, so that the point P,—the

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