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Here, the 1 taken from I, gives 2 for a result, as set down.

ARITHMETICAL COMPLEMENT.

16. The Arithmetical complement of a logarithm is the number which remains after subtracting this logarithm from 10. 10-9.274687 0.725313.

Thus
Hence,

of 9.274687.

0.725313 is the arithmetical complement

17 We will now show that, the difference between two logarithms is truly found, by adding to the first logarithm the arithmetical complement of the logarithm to be subtracted, and then diminishing the sum by 10.

and

Let a the first logarithm

b the logarithm to be subtracted

c=10-b=the arithmetical complement of b. Now the difference between the two logarithms will be expressed by a-b.

But, from the equation c=10-b, we have

c-10=-b

hence, if we place for-b its value, we shall have

a-b=a+c-10

which agrees with the enunciation.

When we wish the arithmetical complement of a logarithm, we may write it directly from the table, by subtracting the left hand figure from 9, then proceeding to the right, subtract each figure from 9 till we reach the last significant figure, which must be taken from 10: this will be the same as taking the logarithm from 10.

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17

Hence, to perform division by means of the arithmetical complement we have the following

RULE.

To the logarithm of the dividend add the arithmetical complement of the logarithm of the divisor: the sum, after subtracting 10, will be the logarithm of the quotient.

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In this example, the sum of the characteristics is 8, from which, taking 10, the remainder is 2.

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1. GEOMETRY is the science which has for its object the measurement of extension.

Extension has three dimensions, length, breadth, height, or thickness.

2 A line is lengh without breadth, or thickness.

The extremities of a line are called points: a point, therefore, has neither length, breadth, nor thickness, but position only.

3. A straight line is the shortest distance from one point to another.

4. Every line which is not straight, or composed of straight lines, is a curved line.

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Thus, AB is a straight line; ACDB is

a broken line, or one composed of straight A lines; and AEB is a curved line.

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The word line, when used alone, will designate a straight line; and the word curve, a curved line.

5. A surface is that which has length and breadth, without height or thickness.

6. A plane is a surface, in which, if two points be assumed at pleasure, and connected by a straight line, that line will lie wholly in the surface.

7. Every surface, which is not a plane surface, or composed of plane surfaces, is a curved surface.

8. A solid or body is that which has length, breadth, and thickness; and therefore combines the three dimensions of extension.

9. When two straight lines, AB,AC, meet each other, their inclination or opening is called an angle, which is greater or less as the lines are more or less inclined or opened. The point of intersection A is the vertex of the angle, and the lines AB, A AC, are its sides.

-B

The angle is sometimes designed simply by the letter at the vertex A; sometimes by the three letters BAC, or CAB, the letter at the vertex being always placed in the middle. Angles, like all other quantities, are susceptible of addition, subtraction, multiplication, and division.

Thus the angle DCE is the sum of the two angles DCB, BCE; and the angle DCB is the difference of the

A

10. When a straight line AB meets another straight line CD, so as to make the adjacent angles BAC, BAD, equal to each other, each of these angles is called a right angle; and the line AB is said to be C perpendicular to CD.

11. Every angle BAC, less than a right angle, is an acute angle; and every angle DEF, greater than a right angle, is an obtuse angle.

E

12. Two lines are said to be parallel, when being situated in the same plane, they cannot meet, how far soever, either way, both of them be produced

13. A plane figure is a plane terminated on all sides by lines, either straight or curved.

If the lines are straight, the space they enclose is called a rectilineal figure, or polygon, and the lines themselves, taken together, form the contour, or perimeter of the polygon.

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14. The polygon of three sides, the simplest of all, is called a triangle; that of four sides, a quadrilateral; that of five, a pentagon; that of six, a hexagon; that of seven, a heptagon; that of eight, an octagon; that of nine a nonagon; that of ten, a decagon; that of twelve, a dodecagon.

C

15. An equilateral triangle is one which has its three sides equal; an isosceles triangle, one which has two of its sides equal; a scalene triangle, one which has its three sides unequal. 16. A right-angled triangle is one which has a right angle. The side opposite the right angle is called the hypothenuse. Thus, in the triangle ABC, right-angled at A, B

17. Among the quadrilaterals, we distinguish :

The square, which has its sides equal, and its angles right angles.

The rectangle, which has its angles right angles, without having its sides equal.

The parallelogram, or rhomboid, which has its opposite sides parallel.

The rhombus, or lozenge, which has its sides equal, without having its angles right angles.

And lastly, the trapezoid, only two of whose sides are parallel.

B

18. A diagonal is a line which joins the vertices of two angles not adjacent to each other. Thus, AF, AE, AD, AC, are diagonals.

19. An axiom is a self-evident proposition.

E

20. A theorem is a truth, which becomes evident by means of a train of reasoning called a demonstration.

21. A problem is a question proposed, which requires a solution.

22. A lemma is a subsidiary truth, employed for the demonstration of a theorem, or the solution of a problem.

23. The common name, proposition, is applied indifferently, to theorems, problems, and lemmas.

24. A corollary is an obvious consequence, deduced from one or several propositions.

25. A scholium is a remark on one or several preceding propositions, which tends to point out their connexion, their use, their restriction, or their extension.

26. A hypothesis is a supposition, made either in the enun

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