PRIMES

, denote the first divisions into which some whole or integer is divided. As, a minute, or Prime minute, the 60th part of a degree; or the first place of decimals, being the 10th parts of units; or the first division of inches in duodecimals, being the 12th parts of inches; &c.

Prime Numbers, are those which can only be measured by unity, or exactly divided without a remainder, 1 being the only aliquot part: as 2, 3, 5, 7, 11, 13, 17, &c. And they are otherwise called Simple, or Incomposite numbers. No even number is a Prime, because every even number is divisible by 2. No number that ends with 0 or 5 is a Prime, the former being divisible by 10, and the latter by 5. The following Table contains all the Prime numbers, and all the odd composite numbers, under 10,000, with the least Prime divisors of these; the description, nature, and use of which, see immediately following the Table.

A Table of Prime and Composite Odd Numbers, under 10,000.
	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16		17	18	19	20	21	22	23	24	25	26	27	28	29	30	31	32	33
01			3	7		3			3	17	7	3			3	19		01	3			3	11	31	3	7	41	3	37		3		7	3
03			7	3	13		3	19	11	3	17		3		23	3	7	03	13	3	11		3		7	3		19	3			3	29		3
07			3		11	3		7	3		19	3	17		3	11		07	3	13		3	7		3	29	23	3		7	3	31	13	3
09	3		11	3			3			3			3	7		3		09		3	23	7	3	47		3	13		3	53		3			3
11		3			3	7	13	3			3	11	7	3	17		3	11	29		3			3			3	7		3	41		3	13	7
13			3		7	3		23	3	11		3		13	3	17		13	3	7		3			3	19	7	3		29	3	23	11	3
17		3	7		3	11		3	19	7	3			3	13	37	3	17	17	23	3		29	3	7		3		11	3		7	3		31
19		7	3	11		3			3			3	23		3	7		19	3	17	19	3	13	7	3	41	11	3			3			3
21	3	11	13	3			3	7		3		19	3		7	3		21		3	17	43	3		1<*>	3			3	7	23	3			3
23		3		17	3		7	3		13	3			3			3	23			3	7	11	3	23		3	43	7	3	37		3	11
27	3			3	7	17	3			3	13	7	3			3		27	11	3	41		3	17	13	3	7	37	3	11		3	53	7	3
29		3		7	3	23	17	3			3			3		11	3	29	7	31	3			3	17	7	3	11		3	29	13	3
31			3			3		17	3	7		3		11	3		7	31	3			3		23	3	11		3		19	3	7	31	3
33	3	7		3		13	3		7	3		11	3	31		3	23	33		3		19	3	7		3	17		3		7	3	13	53	3
37			3		19	3	7	11	3		17	3		7	3	29		37	3	11	13	3			3		43	3	7		3			3	47
39	3			3		7	3			3		17	3	13		3	11	39	37	3	7		3			3		7	3	17		3	43	41	3
41		3		11	3			3	29		3	7	17	3	11	23	3	41		7	3	13		3			3	19		3	17		3	7	13
43		11	3	7		3			3	23	7	3	11	17	3		31	43	3	19	29	3			3	7		3	13		3	17	7	3
47		3	13		3			3	7		3	31	29	3		7	3	47			3	23	19	3			3		41	3	7	11	3	17
49	7		3			3	11	7	3	13		3		19	3		17	49	3	43		3	7	13	3	31		3		7	3		47	3	17
51	3			3	11	19	3		23	3			3	7		3	13	51	17	3		7	3			3		11	3		13	3	23		3
53		3	11		3	7		3			3		7	3			3	53		17	3			3	13	11	3	7		3		43	3		7
57	3			3			3			3	7	13	3	23	31	3		57	7	3	19	11	3	37		3			3			3	7		3
59		3	7		3	13		3		7	3	19		3			3	59		11	3	29	17	3	7		3		31	3	11	7	3
61		7	3	19		3			3	31		3	13		3	7	11	61	3		37	3		7	3	23	13	3	11		3		29	3
63	3			3			3	7		3			3	29	7	3		63	41	3	13		3	31	17	3	11		3	7		3		13	3
67			3			3	23	13	3		11	3	7		3			67	3		7	3	11		3		17	3		47	3			3	7
69	3	13		3	7		3		11	3		7	3	37	13	3		69	29	3	11		3		23	3	7	17	3	19		3		7	3
71		3		7	3		11	3	13		3		31	3			3	71	7		3	19	13	3		7	3		17	3		37	3
73			3		11	3			3	7	29	3	19		3	11	7	73	3			3	41		3		31	3	47	13		7	19	3
77	7	3		13	3			3			3	11		3	7	19	3	77			3	31	7	3			3			3	13	17	3	29	11
79			3			3	7	19	3	11	13	3		7	3		23	79	3			3		43	3	37		3	7		3		11	3	31
81	3			3	13	7	3	11		3	23		3			3	41	81	13	3	7		3			3	29	7	3	43	11	3		17	3
83		3			3	11		3			3	7		3			3	83		7	3		37	3		13	3		11	3	19		3	7	17
87	3	11	7	3			3			3			3	19		3	7	87		3			3		7	3	13		3		29	3		19	3
89		3	17		3	19	13	3	7	23	3	29		3		7	3	89			3		11	3		19	3			3	7		3	11
91	7		3	17		3		7	3			3		13	3	37	19	91	3	31	11	3	7	29	3	47		3		7	3	11		3
93	3			3	17		3	13	19	3			3	7		3		93	11	3		7	3			3			3	11	41	3	31	37	3
97			3		7	3	17		3			3		11	3			97	3	7		3	13		3	11	7	3			3	19	23	3	43
99	3		13	3			3	17	29	3	7	11	3			3		99	7	3			3	11		3	23		3	13		3	7		3

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A Table of Prime and Composite Odd Numbers, under 10,000.
	34	35	36	37	38	39	40	41	42	43	44	45	46	47	48	49	50		51	52	53	54	55	56	57	58	59	60	61	62	63	64	65	66	67
01	19	3	13		3	47		3		11	3	7	43	3		13	3	01		7	3	11		3			3	17		3		37	3	7
03	41	31	3	7		3		11	3	13	7	3			3			03	3	11		3		13	3	7		3	17		3	19	7	3
07		3		11	3			3	7	59	3		17	3	11	7	3	07		41	3			3	13		3		31	3	7	43	3		19
09	7	11	3		13	3	19	7	3	31		3	11	17	3			09	3			3	7	71	3	37	19	3	41	7	3	13	23	3
11	3		23	3	37		3			3	11	13	3	7	17	3		11	19	3	47	7	3	31		3	23		3			3	17	11	3
13		3		47	3	7		3	11	19	3		7	3		17	3	13		13	3		37	3	29		3	7		3	59	11	3	17	7
17	3			3	11		3	23		3	7		3	53		3	29	17	7	3	13		3	41		3	61	11	3			3	7	3	3
19	13	3	7		3			3		7	3		31	3	61		3	19		17	3			3	7	11	3	13	29	3	71	7	3
21	11	7	3	61		3		13	3	29		3			3	7		21	3	23	17	3		7	3		31	3			3			3	11
23	3	13		3			3	7	41	3			3		7	3		23	47	3		11	3		59	3		19	3	7		3	11	37	3
27	23		3		43	3			3		19	3	7	29	3	13	11	27	3		7	3		17	3			3	11	13	3		61	3	7
29	3		19	3	7		3			3	43	7	3		11	3	47	29	23	3	73	61	3	13	17	3	7		3			3		7	3
31	47	3		7	3		29	3		61	3	23	11	3			3	31	7		3			3	11	7	3	37		3	13	59	3	19	58
33			3			3	37		3	7	11	3	41		3		7	33	3			3	11	43	3	19	17	3		23	3	7	47	3
37	7	3		37	3	31	11	3	19		3	13		3	7		3	37	11		3		7	3		13	3		17	3		41	3
39	19		3		11	3	7		3		23	3		7	3	11		39	3	13	19	3	29		3			3	7	17	3	47	13	3	23
41	3		11	3	23	7	3	41		3		19	3	11	47	3	71	41	53	3	7		3			3	13	7	3	79	17	3	31	29	3
43	11	3		19	3		13	3		43	3	7		3	29		3	43	37	7	3		23	3			3			3		17	3	7	11
47	3		7	3			3	11	31	3			3	47	37	3	7	47		3		13	3		7	3	19		3		11	3		17	3
49		3	41	23	3	11		3	7		3			3	13	7	3	49	19	29	3		31	3			3	23	11	3	7		3	61	17
51	7	53	3	11		3		7	3	19		3			3			51	3	59		3	7		3		11	3		7	3			3	43
53	3	11	13	3		59	3			3	61	29	3	7	23	3	31	53		3	53	7	3		11	3			3	13		3			3
57			3	13	7	3			3			3		67	3		13	57	3	7	11	3			3		7	3	47		3	11	79	3	29
59	3			3	17	37	3			3	7	47	3		43	3		59	7	3	23	53	3		13	3	59	73	3	11		3	7		3
61		3	7		3	17	31	3		7	3		59	3		11	3	61	13		3	43	67	3	7		3	11	61	3		7	3
63		7	3	53		3	17	23	3			3		11	3	7	61	63	3	19	31	3		7	3	11	67	3	7		3	23		3
67		3	19		3		7	3	17	11	3		13	3	31		3	67		2<*>	3	7	19	3	73		3			3		29	3	59	67
69		43	3		53	3	13	11	3	17	41	3	7	19	3		37	69	3	11	7	3			3		47	3	31		3			3	7
71	3			3	7	11	3	43		3	17	7	3	13		3	11	71		3	41		3	53	29	3	7	13	3		23	3		7	3
73	23	3		7	3	29		3			3	17		3	11		3	73	7		3	13		3	23	7	3			3			3		13
77	3	7		3		41	3		7	3	11	23	3	17		3		77	31	3	19		3	7	53	3	43	59	3		7	3		11	3
79	7	3	13		3	23		3	11	29	3	19		3	7	13	3	79			3			3			3		37	3		11	3
81	59		3	19		3	7	37	3	13		3	31	7	3	17		81	3			3		13	3			3	7	11	3			3
83	3		29	3	11	7	3	47		3			3		19	3	13	83	71	3	7		3			3	31	7	3	61	13	3	29	41	3
87	11	17	3	7	13	3	61	53	3	41	7	3	43		3			87	3	17		3	37	11	3	7		3	23		3	13	7	3	11
89	3	37	7	3			3	59		3	67	13	3			3	7	89		3	17	11	3		7	3	53		3	19		3	11		3
91		3		17	3	13		3	7		3			3	67	7	3	91	29	11	3	17		3		43	3		41	3	7		3
93	7		3		17	3		7	3	23		3	13		3		11	93	3	67		3	7		3	71	13	3	11	7	3	43	19	3
97	13	3			3	7	17	3			3		7	3	59	19	3	97			3	23	29	3	11		3	7		3		73	3	37	7
99		59	3	29	7	3		13	3	53	11	3	37		3			99	3	7		3	11	41	3	17		3			3	67		3	13

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A Table of Prime and Composite Odd Numbers, under 10,000.
	68	69	70	71	72	73	74	75	76	77	78	79	80	81	82	83		84	85	86	87	88	89	90	91	92	93	94	95	96	97	98	99
01	3	67		3	19	7	3	13	11	3	29		3		59	3	01	31		3	7	13	3		19	3	71	7	3		89	3
03		3	47		3	67	11	3			3	7	53	3	13	19	03	3	11	7	3		29	3			3		13	3	31		3
07	3		7	3			3			3	37		3	11	29	3	07	7	47	3			3		7	3	41	23	3	13	17	3
09	11	3	43		3		31	3	7	13	3	11		3		7	09	3	67		3	23	59	3			3	97	37	3	7	17	3
11	7		3	13		3		7	3	11	73	3			3		11	13	3	79	31	3	7		3	19		3		7	3		11
13	3	31		3		71	3	11	23	3	13	41	3	7	43	3	13	47		3		7	3		13	3	67		3		11	3	23
17	17		3	11	7	3			3			3			3		17	19	3	7	23	3	37	71	3	13	7	3	31	59	3		47
19	3	11		3		13	3	73	19	3	7		3	23		3	19		7	3			3	29	11	3			3			3	7
21	19	3	7		3		41	3		7	3	89	13	3		53	21	3		3	3		11	3	7		3			3		7	3
23		7	3	17	31	3	<*>3		3			3	71		3	7	23		3		11	3		7	3	23		3	89		3	11
27		3			3	17	7	3	29		3		23	3	19	11	27	3			3	7	79	3			3	11	7	3	71	31	3
29		13	3			3	17		3	59		3	7	11	3		29		3		7	3			3	11	19	3	13		3
31	3	29	79	3	7		3	17	13	3	41	7	3	47		3	31		19	3			3	11	23	3	7		3		37	3
33		3	13	7	3			3	17	11	3		29	3		13	33	3	7	89	3	11		3		7	3			3			3
37	3	7	31	3		11	3		7	3	17		3	79		3	37	11		3			3	7		3			3	23	7	3	19
39	7	3		11	3	41	43	3		71	3	17		3	7	31	39	3		53	3		7	3	13		3			3			3
41		11	3	37	13	3	7		3			3	11	7	3	19	41	23	3			3			3			3	7	31	3	13
43	3	53		3		7	3	19		3	11	13	3	17		3	43			3	7	37	3		41	3		7	3			3	61
47	41		3	7		3	11		3	61	7	3	13		3	17	47		3			3	23	83	3	7	13	3		11	3	43	7
49	3		7	3	11		3			3	47		3	29	73	3	49	7	83	3	13		3		7	3		11	3			3
51	13	3	11		3			3	7	23	3		83	3	37	7	51	3	17	41	3	53		3		11	3	13		3	7		3
53	7	17	3	23		3	29	7	3			3		31	3		53	79	3	17		3	7	11	3	19	47	3	41	7	3	59	37
57		3		17	3	7		3	13		3	73	7	3	23	61	57	3	43	11	3	17	13	3			3	7	19	3	11		3
59	19		3		7	3			3		29	3		41	3	13	59	11	3	7	19	3	17		3	47	7	3	11	13	3		23
61	3		23	3	53	17	3		47	3	7	19	3		11	3	61		7	3			3	13		3	11		3		43	3	7
63		3	7	13	3	37	17	3	79	7	3		11	3			63	3			3			3	7	59	3		73	3	13	7	3
67	3		37	3	13	53	3	7	11	3		31	3		7	3	67		13	3	11		3		89	3	17		3	7		3
69		3		67	3		7	3		17	3	13		3			69	3	11		3	7		3	53	13	3	17	7	3		71	3
71			3	71	11	3	31	67	3	19	17	3	7		3	11	71	43	3	13	7	3		47	3	73		3	17	19	3		13
73	3	19	11	3	7	73	3			3		7	3	11		3	73	37		3	31	19	3	43		3	7		3	17	29	3
77	13		3		19	3			3	7		3	41	13	3		77	7	3		67	3	47	29	3			3	61		3	7	11
79	3	7		3	29	47	3	11	7	3		79	3		17	3	79	61	23	3		13	3	7	67	3	83		3		7	3	17
81	7	3	73	43	3	11		3		31	3	23		3	7	1<*>	81	3			3	83	7	3			3	19	11	3		41	3
83			3	11		3	7		3	43		3	59	7	3	83	83	17	3	19		3	13	31	3		11	3	7	23	3		67
87	71	3	19		3	83		3		13	3	7		3			87	3	31	7	3		11	3		19	3	53		3			3
89	83	29	3	7	37	3			3		7	3		19	3		89	13	3		11	3	89	61	3	7	41	3	43		3	11	7
91	3		7	3	23	19	3			3	13	61	3			3	91	7	11	3	59	17	3		7	3			3	11		3	97
93	61	3	41		3		59	3	7		3			3		7	93	3	13		3		17	3	29		3	11	53	3	7	13	3
97	3		47	3		13	3	71	43	3	53	11	3	7		3	97	29		3	19	7	3	11	17	3			3		97	3	13
99		3	31	23	3	7		3		11	3	19	7	3	43	37	99	3			3	11		3		17	3	7	29	3	41	19	3

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Out of the foregoing Table, are omitted all the odd numbers that end with 5, because it is known that 5 is a divisor, or aliquot part of every such number.—— The disposition of the Prime and composite odd numbers in this Table, is along the top line, and down the first or left-hand column; while their least Prime divisors are placed in the angles of meeting in the body of the page. Thus, the figures along the top line, viz, 0, 1, 2, 3, 4, &c, to 99, are so many hundreds; and those down the first column, from 1 to 99 also, are units or ones; and the former of these set before the latter, make up the whole number, whether it be Prime or composite; just like the disposition of the natural numbers in a table of logarithms. So the 16 in the top line, joined with the 19 in the first column, makes the number 1619: the angle of their meeting, viz, of the column under 16, and of the line of 19, being blank, shews that the number 1619 has no aliquot part or divisor, or that it is a Prime number. In like manner, all the other numbers are Primes that have no figure in their angle of meeting, as the numbers 41, 401, 919, &c. But when the two parts of any number have some figure in their angle of meeting, that figure is the least divisor of the number, which is therefore not a Prime, but a composite number: so 301 has 7 for its least divisor, and 803 has 11 for its least divisor, and 1633 has 23 for its least divisor.

Hence, by the foregoing Table, are immediately known at sight all the Prime numbers up to 10,000; and hence also are readily found all the divisors or aliquot parts of the composite numbers, namely in this manner: Find the least divisor of the given number in the Table, as above; divide the given number by this divisor, and consider the quotient as another or new number, of which find the least divisor also in the Table, dividing the said quotient by this last divisor; and so on, dividing always the last quotient by its least divisor found in the Table, till a quotient be found that is a Prime number: then are the said divisors and the last or Prime quotient, all the simple or Prime divisors of the first given number; and if these simple divisors be multiplied together thus, viz, every two, and every three, and every four, &c, of them together, the several products will make up the compound divisors or aliquot parts of the first given number; noting, that if the given number be an even one, divide it by 2 till an odd number come out.

For example, to find all the divisors or component factors of the number 210. This being an even number, dividing it by 2, one of its divisors, gives 105; and this ending with 5, dividing it by 5, another of its factors, gives 21; and the least divisor of 21, by the

2	3	5	7
	6	10	14
		15	21
		30	42
			35
			70
			105
			210

Table is 3, the quotient from which is 7; therefore all the Prime or simple factors of the given number. are 2, 3, 5, 7. Set these therefore down in the first line as in the margin; then multiply the 2 by the 3, and set the product 6 below the 3; next multiply the 5 by all that precede it, viz, 2, 3, 6, and set the products below the 5; lastly multiply the 7 by all the seven factors preceding it, and set the products below the 7; so shall we have all the fac- tors or divisors of the given number 210, which are these, viz, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105.

Prime Vertical, is that vertical circle, or azimuth, which is perpendicular to the meridian, and passes through the east and west points of the horizon.

Prime Verticals, in Dialling, or Prime-Vertical Dials, are those that are projected on the plane of the Prime vertical circle, or on a plane parallel to it. These are otherwise called direct, erect, north, or south dials.

Prime of the Moon, is the new moon at her first appearance, for about 3 days after her change. It means also the Golden Number; which see.

PRIMUM Mobile, in the Ptolomaic Astronomy, is supposed to be a vast sphere, whose centre is that of the world, and in comparison of which the earth is but a point. This they describe as including all other spheres within it, and giving motion to them, turning itself and all the rest quite round in 24 hours.