DECIMAL ARITHMETIC.

MANY personas who have acquired considerable skill in common Arithmetic, are unacquainted with the method of calculating by decimals, which is of great use in Navigation; for which reason it was thought proper to prefix the following brief explanation.

Fractions or Vulgar Fractions are expressions for any assignable part of an unit; they are usually denoted by two numbers, placed the one above the other, with a line between them: thus, & denotes the fraction one-fourth, or one part out of four of some whole quantity, considered as divisible into four equal parts. The lower number 4 is called the denominator of the fraction, showing into how many parts the whole or integer is divided; and the upper number 1, is called the numerator, and shows how many of those equal parts are contained in the fraction. And it is evident that if the numerator and denominator be varied in the same ratio, the value of the fraction will remain unaltered: thus if the numerator and denominator of the fraction be multiplied by 2, 3, or 4, &c. the fractions arising will be,,, &c, which are evidently equal to .

Decimal Fraction is a fraction whose denominator is always an unit with some number of ciphers annexed, the numerators of which may be any numbers whatever; as,,, &c. And as the denominator of a decimal is always one of the numbers 10, 100, 1000, &c. the inconvenience of writing these denominators may be avoided, by placing a point between the integral and the fractional part of the number; thus is written .3; and is written .14; the mixed number 31, consisting of whole numbers and fractional ones is written 3.14.

In setting down a decimal fraction, the numerator must consist of as many places as there are ciphers in the denominator; and if it has not so many figures the defect must be supplied by placing ciphers before them; thus, 16 16.016, 0.0016, &c. And as ciphers on the right hand side of integers increase their value in a tenfold proportion, as 2, 20, 200, &c. so when set on the left hand of decimal fractions, they decrease their value in a tenfold proportion, as 2, .02, .002, &c. but ciphers set on the right hand of these fractions make no alteration in their value, neither of increase or decrease; thus, .2 is the same as .20 or .200. The common arithmetical operations are performed the same way in decimals, as they are integers; regard being had only to the particular notation, to distinguish the integral from the fractional part of a sum.

ADDITION OF DECIMALS.

Addition of decimals is performed exactly like that of whole numbers, placing the numbers of the same denomination under each other, in which case the decimal separating points will range straight in one column.

SUBTRACTION OF DECIMALS.

Subtraction of decimals is performed in the same manner as in whole numbers, by observing to set the figures of the same denomination and the separating points directly under each other.

MULTIPLICATION OF DECIMALS.

Multiply the numbers together the same as if they were whole numbers, and point off as many decimals from the right hand as there are decimals in both factors together; and when it happens that there are not so many figures in the product as there must be decimals, then prefix as many ciphers to the left hand as will supply the defect.

Division of decimals is performed in the same manner as in whole numbers; only observing that the number of decimals in the quotient must be equal to the excess of the number of decimals of the dividend above those of the divisor.-When the divisor contains more decimals than the dividend, ciphers must be affixed to the right hand of the latter to make the number equal or exceed that of the divisor.

REDUCTION OF DECIMALS.

If you wish to reduce a vulgar fraction to a decimal, you may add any number of ciphers to the numerator, and divide it by the denominator, the quotient will be the decimal fraction; the decimal point must be so placed that there may be as many figures to the right hand of it as you added ciphers to the numerator; if there are not as many figures in the quotient, you must place ciphers to the left hand to make up the number.

If you have any decimal fraction, it is easy to find its value in the lower denominations of the same quantity; thus if the fraction was the decimal of a yard, by multiplying it by 3 we have its value in feet and parts; if we multiply this by 12, the product is its value in inches and parts; and in the same manner the values may be obtained in other cases.

GEOMETRY.

GEOMETRY is the Science which treats of the description, properties, and relations of magnitudes in general, of which there are three kinds or species, viz. a line which has only length without either breadth or thickness; a superfices, comprehended by length and breadth, and a solid, which has length, breadth, and thickness.

I.
II.

A POINT considered mathematically has no length, breadth, or thickness.

A STRAIGHT LINE OR RIGHT LINE is the shortest distance between the two points which limits its length, as AC III.

A PLANE SUPERFICES is that in which any two points being taken, the straight line between them lies wholly in that surface.

IV.

PARALLEL LINES are such as are in the same plane and which extended infinitely do never meet, as AB, DC.

V.

A CIRCLE is a plane figure, bounded by an uniform curve line; it is commonly described with a pair of compasses; one point of which is fixed, whilst the other is turned round to the place where the motion first began; the fixed point is called the CENTRE, and the line described by the other point is called the CIRCUMFERENCE. G

VI.

The RADIUS of a circle, or SEMIDIAMETER, is a right line drawn from the centre to the circumference, as AC; or it is that line which is taken between the points of the compasses to describe the circle. A

A DIAMETER of a circle is a right line drawn through the centre and terminated at both ends by the circumference, as ACB, and is the double of the radius AC. A diameter divides the cirele, and its circumference into two equal parts.

VII.

VIII.

Ꭰ

F

An ARCH of a circle is any part or portion of the circumference, as DFE.

B

E

The CHORD of an arch is a straight line joining the ends of the arch; it divides the circle into two unequal parts, called SEGMENTS, and is a chord to them both, as DE is the chord of the arches DFE and DGE.

IX.

A SEMICIRCLE, or half circle, is a figure contained under a diameter and the arch terminated by that diameter, as AGB or AFB. Any part of a circle contained between two radii and an arch, is called a SECTOR.

X.

A QUADRANT is half a semicircle, or one-fourth part of a whole circle, as the figure CAG.

NOTE. All circles, whether great or small, are supposed to have their circumference divided into 360 equal parts, called degrees, and each degree into 60 equal parts, called minutes; and each minute into 60 equal parts, called seconds, and so on into thirds, fourths, &c. and an arch is said to be of as many degrees as it contains parts of the 360, into which the circumference is divided.

A new division of the circumference of the circle has lately been adopted by several eminent French mathematicians, in which the quadrant is divided into 100°, each degree into 100, each minute into 100", &c. and tables of logarithms have been published conformable thereto. The general adoption of this division would tend greatly to facilitate most of the calculations of navigation and astronomy.

An angle is usually expressed by the letter placed at the angular point, as the angle A. But when two or more angles are at the same point, it is then necessary to express each by three letters, and the letter at the angular point is placed between the two. Thus, the angle formed by the lines AB, AC, is call-A

ed the angle BAC or CAB, and that formed by AB, AD, is called the angle BAD, or DAB.

An angle is measured by the arch of a circle comprehended between the two

legs that form the angle, the centre of the circle being the angular point.
Thus the angle A is measured by the arch BC describ-
ed round the point A as a centre, and the angle is said to
be of as many degrees as the arch is, that is, if the arch
BC is 30°, then the angle BAC, is said to be an angle of 30 A

degrees.

XII.

A

B

C

If a right line AB, fall upon another DC, so as to incline neither to the one side nor the other, but makes the angles ABC, ABD, equal to each other; then the line AB is said to be perpendicular to the line DC, and each of these angles is called a right angle, being each equal to a quadrant or 90°; because the sum of the two angles ABC, ABD, is measured by the semicircle DAC, described on the diameter DBC, and centre B.

XIII.

An ACUTE ANGLE is less than a right angle, as ABC.

C

B

D

A RIGHT-ANGLED TRIANGLE has one of its angles right; the side opposite the right angle is called the hypotenuse; and the other two sides are called legs; that which stands upright, is called the perpendicular, and the other the base; thus BC is the hypotenuse, AC the perpendicular, and AB the base; the angles opposite the two legs are both acute.