It will be noticed that in multiplication the order of the figures or letters will be immaterial, as 7 × 5 will equal 5 × 7; but in using the sign of Division, it must be borne in mind that the first number is to be divided by the other; as 158 would give 15 eighths, whilst 8-15 would only give 8 fifteenths.

The sign = means equal. Thus, 2+3 = 5. It may also be used to show that two surfaces are equivalent or equal in area; thus, the square of 4 inches, being 16 inches, will be equivalent with a rectangle 8 by 2; and in geometrical operations the symbol of equality may be used, its limited application being obvious, and its application being desirable, to shorten the statement of the deduction.

The sign> or < denotes inequality, and designates the greater or the less of two numbers, lines, &c., the angular point turning towards the less; thus, A > B means that A is greater than B,' whilst A < B means that A is less than B.

The sign, or the letter S placed horizontally, means "similar to."

23. Proportions are expressed by colons and double colons as to geometrical proportions, with which geometry has, with rare exceptions, to deal; or by dots and double dots in relation to arithmetical progression; thus :—

Geometrical progression, 3:59:15, means, as 3 to 5,

so will be 9 to 15.

Arithmetical progression, 3. 5: 9. 11, the simple difference, and not the multiple or division, giving the result.

24. The sign denotes the extraction of roots, a

figure being placed over it for other than square roots; thus, √ 4, means the square root of 4, being 2, √38, the cube root of 8, being 2, ✓16, the 4th root of 16, being 2, and so on.

Where the root of several numbers or letters is to be taken, the sign will be continued by a line over the numbers or letters, or they will be included in ( ). Thus, √4 × 12 or √ (4 × 12) means the square root of 4 × 12, or 48, or ✔A.B, the square root of A and B multiplied together.

25. Powers are marked by small figures above the figure or letter; thus, 62 means the square of 6 (36), 63 the cube of 6 (216), and so on; or A3, A3, &c.

Where the sum of several numbers, or the product or difference, is to be raised to the required power, they are joined; thus, (6 x 7 -2)2 means the square of 6 × 7, with 2 deducted from the product, or the square of 42-2, or 40.

If (6 × (7 − 2))2 required it would be the square of 6 × 5,

or 30.

26. Combined and Separate Operations. The last example draws attention to the distinction between the separate and combined operations with figures, &c. Where the figures or letters are not denoted as being combined, the operations will be independent, as 6×7-2 means that 6 is to be multiplied by 7, and after that operation is performed, 2 must be subtracted; but with (7-2), or 7 and 2 combined, the 6 must be multiplied by the result of that combination, being 5.

This is exemplified by the common catching question, which is most, twice twenty-four or twice four and twenty; the result in the former case being 48, and in the latter only 28, being twice 4 + 20, and not plus twice twenty.

The sign means because, and.. means therefore.

27. Decimal Points are best marked at the top of figures; thus 8 for decimal 8, though they are sometimes marked thus .8, and occasionally by a comma, thus,8.

Additional Symbols sometimes used:-(a) | right line, (b) || parallel to, (c) ||' parallels, (d) perpendicular to, (e) angle, (ƒ) ▲ triangle, (g) □ parallelogram, (h) O circle, (i) Oce circumference.

In this work the corresponding parallels in the various steps of an operation are marked thus, 1 and 1' for the first pair, 2 and 2' for the second pair, &c.

28. The following terms are also used in geometry, and are mainly stated from an old but most valuable work, in which great care is taken to give explanations fully and clearly (Ward's Young Mathematician's Guide).

Proposition, anything stated for consideration or action, and includes theorems and problems.

A Theorem proposes something to be proved or demon

strated.

A Problem denotes something to be done or constructed. Demonstration is understood to mean the highest degree of proof that human reason is capable of attaining to, by a train of arguments deduced or drawn from such plain axioms and other self-evident truths as cannot be denied by anyone that justly considers them.

Corollary, some consequent truth, drawn or gained from any demonstration.

Lemma, the demonstration of some premises laid down or proposed as a preparation to shorten the proof of a theorem under consideration.

Scholium, a brief commentary or observation made upon some precedent discourse.

AXIOMS.

29. The words are taken from the English translation of Euclid, but as regards No. 12, it is given as an axiom, to cut a knot which Euclid would not take the trouble to untie, nor of which have I seen any satisfactory proof in any English or French work on Geometry.

By applying the principle of common sense to the matter, instead of the hairsplitting distinctions and positions in the attempts at demonstration I have read, I give in Sect. 30 of this Division, and in Fig. A 3, a demonstration which will, I shink, be clearly understood, and the proof made apparent any student.

to

Axiom 1. Things which are equal to the same thing, are equal to one another (A = B. C = B... C = A).

Axiom 2. If equals be added to equals, the wholes are equal (A + B = C + B).

Axiom 3. If equals be taken from equals, the remainders are equal (A-B-C-B).

Axiom 4. If equals be added to unequals the wholes are unequal (Bunequal to D. Then A + B unequal to C + D). Axiom 5. If equals be taken from unequals, the remainders are unequal (B-A unequal to D-C).

Axiom 6. Things which are double of the same are equal to one another (A = 2B. C = 2B .. A = C).

Axiom 7. Things which are halves of the same are equal to one another (A = }B. C = {B .. A = C).

Axiom 8. Magnitudes which coincide with one another, that is which exactly fill the same space, are equal to one another. (For example, squares with 1-inch sides.)

Axiom 9. The whole is greater than its parts. (A > A, &c.)

Axiom 10. Two straight lines cannot enclose a space. (They include a space, but do not enclose it, a third line being required.)

Axiom 11. All right angles are equal to one another. (This would be understood without making it an axiom.)

Axiom 12. (See remark above.) If a straight line meets two straight lines, so as to make the two interior angles on the same side of it taken together less than two right angles, those straight lines being continually produced shall at length meet upon that side in which are the angles which are less than two right angles. (Fig. A 3, in which line AB cuts the lines CD and EF, the angles on the side towards D and F being less than two right angles.)

30. Demonstration of the proposition stated by Euclid

as Axiom 12.

In A Fig. 3, let AB be a straight line meeting 2 straight lines, CD and EF, and making the two interior angles on the same side, a'a F and aa D, together less than 2 right angles.

Then, for the demonstration, through any point in line