NUMBER

, a collection or assemblage of several units, or several things of the same kind; as 2, 3, 4, &c, exclusive of the number 1: which is Euclid's definition of Number.—Stevinus defines Number as that by which the quantity of anything is expressed: agreeably to which Newton conceives a Number to consist, not in a multitude of units, as Euclid defines it, but in the abstract ratio of a quantity of any kind to another quantity of the same kind, which is accounted as unity: and in this sense, including all these three species of Number, viz, Integers, Fractions, and Surds.

Wolfius defines Number to be something which refers to unity, as one right line refers to another. Thus, assuming a right line for unity, a Number may likewise be expressed by a right line. And in this way also Des Cartes considers numbers as expressed by lines, where he treats of the arithmetical operations as performed by lines, in the beginning of his Geometry.

For the manner of characterizing Numbers, see NOTATION. And

For reading and expressing Numbers in combination, see Numeration.

Mathematicians consider Number under a great many circumstances, and different relations, accidents, &c.

Numbers

, Absolute, Abstract, Abundant, Amicable, Applicate, Binary, Cardinal, Circular, Composite, Concrete, Defective, Fractional, Homogeneal, Irrational or Surd, Linear or Mixt, Ordinal, Polygonal, Prime, Pyramidal, Rational, Similar, &c, see the respective adjectives.

Broken Numbers, or Fractions, are certain parts of unity, or of some other Number.

Cubic Number, is the product of a square Number multiplied by its root, or the continual product of a Number twice multiplied by itself; as the Numbers - - 1, 8, 27, 64, 125, &c, which are the cubes of - 1, 2, 3, 4, 5, &c.

This series of the cubes of the ordinal Numbers, may be raised by addition only, viz, adding always the differences; as was first shewn by Peletarius, at the end of his Algebra, first printed in 1558, where he gives a table of the squares and Cubes of the first 140 numbers. See Cube.

Every Cubic Number whose root is less than 6, viz, the Cubic Numbers 1, 8, 27, 64, 125, being divided by 6, the remainder is the root itself:| Thus, ; where the remainders, or the numerators of the small fractions, are 0, 1, 2, 3, 4, 5, the same as the roots of the Cubes 0, 1, 8, 27, 64, 125. After these, the next six Cubic Numbers being divided by 6, the remainders will be respectively the same arithmetical series, viz - - - 0, 1, 2, 3, 4, 5; to each of which adding 6, gives 6, 7, 8, 9, 10, 11, for the roots of the next six cubes 216, 343, &c.

Then, again dividing the next set of six Cubic Numbers, viz, - - 1728, 2197, &c, by 6, the remainders are again the same series, viz, 0, 1, 2, 3, 4, 5, to each of which adding 12, gives 12, 13, 14 15, 16, 17, for the roots of the said next six cubes. And so on in infinitum, the series of remainders 0, 1, 2, 3, 4, 5, continually recurring, and to each set of these remainders the respective Numbers 0, 6, 12, 18, 24, &c, being added, the sums will be the whole series of roots, 0, 1, 2, 3, 4, 5, 6, &c.

M. de la Hire, from considering this property of the Number 6, with regard to Cubic Numbers, found that all other Numbers, raised to any power whatever, had each their divisor, which had the same effect with regard to them, that 6 has with regard to Cubes. And the general rule he has discovered is this: if the exponent of the power of a number be even, i. e. if that number be raised to the 2d, 4th, 6th, &c power, it must be divided by 2, then the remainder added to 2, or to a multiple of 2, gives the root of the Number corresponding to its power, i. e. the 2d, or 4th, &c, root. But if the exponent of the power of the Number be uneven, viz the 3d, 5th, 7th, &c power, the double of that exponent shall be the divisor, which shall have the property here required.

A Determinate Number, is that which is referred to some given unit; as a ternary or three.

An Even Number, is that which may be divided into two equal parts, without remainder or fraction, as the Numbers 2, 4, 6, 8, 10, &c.—The sums, differences, products, and powers of Even Numbers, are also Even Numbers.

An Evenly-Even Number, is such as being divided by an even Number, the quotient is also an Even Number without a remainder: as 16, which divided by 8 gives 2 for the quotient.

An Unevenly-Even Number, is such as being divided by an Even Number, the quotient is an Uneven one: as 20, which divided by 4, gives 5 for the quotient.

Figurate or Figural Numbers, are certain ranks of Numbers found by adding together first a rank of units, which is the first order, which gives the 2d order; then these added give the 3d order; and so on. Hence, the several orders of Eigurate Numbers, are as follow:

First order1 . 1 . 1  . 1  . 1 . &c.
2d order1 . 2 . 3  . 4  . 5 . &c.
3d order1 . 3 . 6  . 10. 15. &c.
4th order1 . 4 . 10. 20. 35. &c.
5th order1 . 5 . 15. 35. 70. &c.

The first order consists all of equals, and the 2d order of the natural arithmetical progression; the 3d order is also called triangular Numbers, the 4th order pyramidals, &c.

See Figurate Numbers.

Heterogeneal Numbers, are such as are referred to different units. As three men and 4 trees.

Homogeneal Numbers, are such as are referred to the same unit. As 3 men and 4 men.

Impersect Numbers, are those whose aliquot parts added together, make either more or less than the whole of the number itself; and are distinguished into Abundant and Defective.

Indeterminate Number, is that which is referred to unity in the general; which is what we call Quantity.

Irrational or Surd Number, is one that is not commensurable with unity; as √2, or √34, &c.

Perfect Number, that which is just equal to the sum of its aliquot parts, added together. As, 6, 28, &c: for the aliquot parts of 6 are 1, 2, 3, whose sum is the same 6; and the aliquot parts of 28, are 1, 2, 4, 7, 14, whose sum is 28. See Perfect Number.

Plane Number, that which arises from the multiplication of two other Numbers: so 6 is a plane or rectangle, whose two sides are 2 and 3, for .

Square Number, is a Number produced by multiplying any given Number by itself; as the Square Numbers - - 1, 4, 9, 16, 25, &c, produced from the roots - 1, 2, 3, 4, 5, &c.

Every Square Number added to its root makes an even Number. See Square.

Uneven Number, or Odd Number, that which differs from an even Number by one, or which cannot be divided into two equal integer parts; such as 1, 3, 5, 7, &c. The sums and differences of Uneven Numbers are even; but all the products and powers of them are Uneven Numbers. On the other hand, the sum or difference of an even and Uneven Number are both Uneven, but their product is even.

Whole Number, or Integer, is unit, or a collection of units.

Golden Number. See Golden Number and Cycle.

Number of Direction, in Chronology, some one of the 35 Numbers between the Easter limits, or between the earliest and latest day on which it can fall, i. e. between March 22 and April 25, which are 35 days; being so called, because it serves as a Direction for finding Easter for any year; being indeed the Number that expresses how many days after March 21, Easterday falls. Thus, Easter-day falling as in the first line below, the Number of Direction will be as on the lower line:

MarchApril
Easter-day, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 1, 2, &c. N<*> of Dir. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, &c and so on, till the Number of Direction on the lower line be 35, which will answer to April 25, being the latest that Easter can happen. Therefore add 21 to the Number of Direction, and the sum will be so many days in March for the Easter-day: if the sum exceed 31, the excess will be the day of April.

To find the Number of Direction. Enter the following table (which is adapted to the New Style), with the Dominical Letter on the left hand, and the Golden Number at the top, then where the columns meet is| the Number of Direction for that year. See Ferguson's Astron. pa. 381, ed. 8vo.

G. N.12345678910111213141516171819
Do<*> Le<*>.
A2919<*>261233191226195261252612331912
B2713627133420132720627136201334206
C8472114352172821728147211428217
D191582215292282915829151221529228
E30162231630239301692<*>16223930239
F2417324103124103117102<*>1732410311710
G251842511321811321842<*>18425113218<*>1

Thus, for the year 1790, the Dominical Letter being C, and the Golden Number 5; on the line of C, and below 5, is 14 for the Number of Direction. To this add 21, the sum is 35 days from the 1st of March, which, deducting the 31 days of March, leaves 4 for the day of April, for Easter-day that year.

Numeral Characters. See Characters.

Numeral Figures. The antiquity of these in England has, for several reasons, been supposed as high as the eleventh century; in France about the middle of the tenth century; having been introduced into both countries from Spain, where they had been brought by the Moors or Saracens. See Wallis's Algebra, pa. 9 &c, and pa. 153 of additions at the end of the same. See also Philos. Trans. numb. 439 and 475.

Numeral Letters, those letters of the alphabet that are commonly used for figures or numbers, as I, V, X, L, C, D, M.

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Entry taken from A Mathematical and Philosophical Dictionary, by Charles Hutton, 1796.

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NOTATION
NOTES
NOVEMBER
NUCLEUS
NUEL
* NUMBER
NUMERATION
NUMERATOR
NUMERICAL