Ans. 19.105.

Ans. .9911.

6. Required the 5th root of .9563. 7. Required the 4th root of .00079. Ans. .16765.

Of the arithmetical complements of logarithms.

When it is required to subtract several logarithms from others, it will be more convenient to convert the subtraction into an addition, by writing down, instead of the logarithms to be subtracted, what each of them wants of 10.00000, which may readily be done, by writing down what the first figure, on the right hand, wants of 10, and what every other figure wants of 9; this remainder is called the Arithmetical Complement. Thus, if the logarithm be 2.53061, its arithmetical complement will be 7.46939. If one or more figures to the right hand be ciphers, write ciphers in their place, and take the first significant figure from 10, and the remaining figures from 9. Thus, if the logarithm be 4.61300, its arithmetical complement will be 5.38700.

In any operation, where the arithmetical complements of logarithms are added to other logarithms, there must be as many 10's subtracted from the sum, as there are arithmetical complements used.

As an example, let it be required to divide the product of 76.4 and 35.84, by the product of 473.9 and 4.76.

GEOMETRY,

DEFINITIONS.

1. GEOMETRY is that science wherein the properties of magnitude are considered.

2. A point is that which has position, but not magnitude.

3. A line has length but not breadth.

4. A straight, or right line, is the shortest line that can be drawn between any two points.

4. A superficies or surface has length and breadth, but not thickness.

6. A plane superficies is that in which any two points being taken, the straight line which joins them lies wholly in that superficies.

7. A plane rectilineal angle is the inclination of twe straight lines to one another, which meet together, but are not in the same straight line, as A, Fig. 1.

Note. When several angles are formed about the samé point, as at B, Fig. 2, each particular angle is expressed by three letters, whereof the middle letter shows the angular point, and the other two, the lines that form the angle; thus, CBD or D B C signifies the angle form

8. The magnitude of an angle depends on the incliation which the lines that form it have to each other, and not on the length of those lines. Thus the angle DBE is greater than the angle A B C, Fig. 3.

9 When a straight line C D stands on another straight line A B, so as to incline to neither side, but makes the angles on each side equal, then those angles A D C and BDC are called right angles, and the line C D is said to be perpendicular to A B, Fig. 4.

10. An acute angle is that which is less than a right an gle, as BD F, Fig. 4.

11. An obtuse angle is that which is greater than a right angle, as AD E, Fig. 4.

12. Parallel straight lines are those which are in the same plane, and which, being produced ever so far both ways, do not meet, as A B, C D, Fig. 5.

13. A figure is a space bounded by one or more lines.

14. A plane triangle is a figure bounded by three straight lines, as ABC, Fig. 6.

15. An equilateral triangle has its three sides equal to each other, as A, Fig. 7.

16. An isosceles triangle has only two of its sides equal, as B, Fig. 8.

17. A scalene triangle has three unequal sides, as ABC, Fig: 6.

18. A right angled triangle has one right angle, as ABC, Fig. 9: in which the side AC opposite to the right angle is called the hypothenuse.

19. An obtuse angled triangle has one obtuse angle, as C, Fig. 10.

20. An acute angled triangle has all its angles acute, as ABC, Fig. 6,

21. Acute and obtuse angled triangles are called oblique angled triangles.

22. Any plane figure bounded by four right lines, is called a quadrilateral.

23. Any quadrilateral, whose opposite sides are paral lel, is called a parallelogram, as D, Fig. 11.

24. A parallelogram, whose angles are all right angles, is called a rectangle, as E, Fig. 12.

25. A parallelogram whose sides are all equal, and angies right, is called a square, as F, Fig. 13.

26. A rhomboides is a parallelogram, whose opposite sides are equal angles oblique, as D, Fig. 11.

27. A rhombus is a parallelogram, whose sides are all equal and angles oblique, as G, Fig. 14.

28. Any quadrilateral figure that is not a parallelogram, is called a trepezium.

29. A right line joining any two opposite angles of a quadrilateral figure, is called a diagonal.

30. That side AB upon which any parallelogram ABE C, or triangle A B C is supposed to stand, is called the base; and the perpendicular CD falling thereon from the opposite angle C, is called the altitude of the