EXAMPLE.

Subtract an isosceles triangle whose sides 2 inches each have a base of 1 inch, from an equilateral triangle whose base is 2 inches.

PROBLEM LXXX.

To make a square equal to the difference of two squares.

1. Let O G be the side of the greater of the given squares. Vide Prob. LXVI..

2. From O as a centre, with O G as a radius describe an arc GH. On O G take O F equal to the side of the second square. From F draw F H perpendicular to O G, meeting the arc in H; when F H will be the side of the square required. For reason, vide Prob. LXXVIII.

EXAMPLE.

Make a square equal to the difference of two squares whose sides are 2 and three inches.

PROBLEM LXXXI.

To subtract any given quadrilateral figure from any other given figure.

Make a triangle equal to each of the given figures: subtract one triangle from the other, and the remainder will be the difference required.

PROBLEM LXXXII.

Two triangles being given, to make a square equal to their difference.

Transform the given triangles to squares; and make a square equal to their difference, by Prob. LXXX.

PROBLEM LXXXIII.

To make a triangle equal to a common ellipse.

1. Let ADEB GF be the given D ellipse.

2. By Problem LXXIV. make triangles equal to the sectors OFAD, VEBG, FIG, and D KE. By Prob. LXXVI. make a triangle equal to the sum of these triangles.

E

B

3. By Prob LX. make a triangle equal to the rhombus OIVK. Subtract it from the above sum, and the remainder will be equal to the triangle required.

The reason will be evident if Probs. LXX. and LXXIV. be understood.

Other exercises in Subtraction can be selected from the Problems in Transformation.

MULTIPLICATION.

PROBLEM LXXXIV.

To multiply a triangle by any given number.

1. Let AD C, Prob. LXXVI. be the given triangle.

2. Produce the base AD to B making AB the required multiple of AD. Join CB; when ACB will be the triangle required.

The triangle ACB is three times the area of ACD, because AB is three times the length of AD.

The reason for this and the following problem is, because "Triangles (and parallelograms) of the same altitudes are to one another as their bases." Vide Prob. I. Bk. VI. Euclid.

EXAMPLE.

Make a right-angled triangle four times the superficial content of an oblique-angled triangle.

PROBLEM LXXXV.

To make a rectangle equal to any multiple of a given figure.

If the given figure be not a parallelogram, transform it to one make a rectangle whose base shall be equal to the required multiple of the base of the parallelogram, and its height equal to the parallelogram, and the work is done.

EXAMPLE.

Construct a rectangle five times the area of an oblong I by an inch.

For other Problems in Multiplication, vide Addition.

DIVISION.

PROBLEM LXXXVI.

To divide a triangle into any number of equal triangles.

Divide either side of the given triangle into the required number of equal parts, join the points of division to the opposite vertex, and the work is done.

From this it will be seen that Division is the reverse of Multiplication.

For reason, vide multiplication.

EXAMPLES.

1. Divide an equilateral triangle whose base is one inch, into five equal parts.

2. Show the fifth part (dividing either of its shorter sides) of any triangle, one of whose angles contains 155 degrees.

PROBLEM LXXXVII.

To divide any triangle into two triangles having any given ratios.

1. Let A CB, Prob. LXXVI. be the given triangle, and let it be required to divide it into any two proportions, (say as one to two.)

2. Add the given ratios together, and call A B their sum.

3. From A cut off A D equal to one of the ratios (say one). Draw CD, which shall divide the triangle ACB into the ratios required, for A C D is the half of D ČB.

EXAMPLE.

Divide any right-angled triangle into two parts, which shall be to each other, as 2 to 74. Vide Prob. XV.

PROBLEM LXXXVIII.

To divide any parallelogram into any given number of parallelograms.

Let ABCD be the given figure : divide the base AB into the given number of equal parts (say four): draw lines from these points of division, parallel to AD, and cutting DC; when the parallelogram shall be divided as required.

D

A.

B

ANOTHER METHOD.

When the given number is even, proceed as in the Fig. to Prob. LVI. and the work is done.

Should the number be an odd one, (and the product of any two other numbers) as nine or fifteen, the process is equally applicable.

EXAMPLE.

Divide a square of 2 inches each side, into 16 equal squares, and draw diagonals therein.

PROBLEM LXXXIX.

To divide any parallelogram into eight equal triangles.

Draw diagonals in the parallelogram, which will divide it into four equal triangles: bisect each of these by Prob. LXXXVI. and the work is done.

For other problems in Division, vide Subtraction.

When the preceding exercises are understood, the ingenious paper puzzles of the day will be very simple.

PROBLEM XC.

To make a square equal to any given proportion of a given square as 1,, . &c.

2 3 4

1. Let E G, Prob. LXVI. be the side of the given square: describe a semicircle on E G. Make GF the given proportion

to G E, say one-fourth.

2. Draw FH perpendicular to EF: join HG; which will be the side of the square required.

EXAMPLE.

Construct a square 2 inches each side, and another ths of

its area.

ON MATHEMATICAL DRAWING.

The preceding Problems form the basis of every branch of drawing, whether it be civil, military, or naval architecture; any department of machinery; or illustrations of science and philosophy. To apply these problems, the student must make himself familiar with decimal fractions; because upon his knowledge of these, will depend the truth of the dimensions of his drawings, for all measures must be expressed decimally, as 25 for 1,5 for an inch, and so on. If he be not expert in arithmetic, he should study it from Professor Christie's valuable treatise, or he may apply to a very cheap and useful work, published by Mr. Black, of Woolwich, being a new edition of Walkingame, and containing the principal modern improve

ments.

The square root, also, should be well understood, as it will meterially assist the practical man in the demonstration of those problems which require more than the first book of Euclid, as may be seen when Prob. XC is drawn and its proof attempted.

A knowledge of Trigonometry is likewise essential for the construction of a triangle, when its angles are given in degrees, as it is very unsafe to depend upon a common protractor to make an angle whose legs are to exceed 3 inches in length. The student is recommended to employ a master to teach him to construct and apply the trigonometrical canon, and especially the line of chords; and herein the principle of the sector, (vide page 23) will be beautifully illustrated.

The student's next step must be to construct Plain and Diagonal Scales, being lines of equal parts, having a given ratio to the English inch, as scales of 1 inch to a foot, 2 inches to a foot, 10 feet to an inch, 5 chains to an inch, and so on. By referring to his ivory protractor, he will find that the divisions of the Diagonal scales on it are much more delicate than those of the Plain Scales, so called. He should also commence making neat, clear drawings with lead pencil or with pen and ink by hand, (called "Rough Sketches,") of buildings, machinery, &c., writing the dimensions on each part. These will serve as guides to enable him to draw his subject accurately to any given scale.

The practice of drawing scales will involve the construction of the Vernier, so called from its being the invention of Peter Vernier, a French gentleman, whose discovery in this contrivance has entitled him to the gratitude of every Military and Civil Engineer.

G