ent is 4. And the oppofite of this Rea fon is Submanifold Superparticular; as 2 to 5, 2 to 7, 3 to7, 4 to 9, &c.

11. Manifold Superpartient Reafon ] is, when the Antecedent contains the Consequent divers times; and divers parts of the Confequent befides; as 8 to 3, 17 to 5, 19 to 4, 28 to 5, c. Here the Quotient of the Antecedent divided by the Confequent, is a Mixt Number, whose whole part, as also the Numerator of the Fraction annexed unto it, is always a number compofed of more Unites than one : So 8 divided by 3, the quotient is 2, and 28 by 5, the quotient is 53, &c. —And the oppofite hereof is Submanifold Superpartition; as 3 to 8, 5 to 17, 4t0 19, 5 to 28, and the like.

These are the several Varieties of the Rates or Reasons that are found among Numbers; fo that no two Numbers can be named, but the Rate or Reafon between them is comprehended under one of thefe laft Five Heads.

CHAP.

III.

of Arithmetical, Geometrical, and Mufical

A

Proportions.

I. Of Arithmetical Proportion.

Rithmetical Proportion (or Habitude) is an equality of Differences: That is to fay, when several Numbers have one and the fame Difference: And this Habitude is twofold, viz. (1.) Continued. (2.) Difcontinued.

1. Arithmetical Proportion Continued ], is, when of feveral Numbers, the Second exceedeth the Firft, by the fame number of Unites, as the Third exceeds the Second, and as the Fourth doth the Third; and fo in infinitum.

As thus,

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, &c. do differ from each other by (1), or One Unite. And 1, 3, 5, 7, 9, &c. differ by (2) or Two Unites. Alfo 1, 6, 11, 16, 21, 26, &c. differ by (5) or Five Unites. And this Orderly proceeding of Numbers from the leffer to the greater, is that which we properly call Arithmetical Progreffion.

2. Arithmetical Proportion Difcontinued ], is, when the Second Number exceeds the First, by the fame number of Unites as the Fourth doth the Third; but not as the Third doth the Second.

As for Example;

1, 3, 7, 9, are four diftinct Arithmetical Proportionals Difcontinued. For 3 exceeds 1 by the fame number of Unites (viz. 2.) as 9 exceeds 7, but not as 7 doth 3. So again, 2, 7, 10, 15, are four Arithmetical Proportionals Difcontinued; for 7 exceedeth 2 by (5), and fo doth 15 exceed 10; but 10 doth not exceed 7 by (5), but by (3), And this is Arithmetical Proportion Discontinued.

II. Of Geometrical Proportion.

Geometrical Proportion, or Habitude,] is the equality of Ratio's; or it is that which fhews what part or parts one Number is of another Or more plainly thus; It is an Increase by a Common Multipli

cation.

So these Numbers, 1, 2, 4, 8, 16, 32, 64, &c. are Numbers in Geometrical Proportion. where the Common Multiplier is [2], that is, the first multiplied by 2, produceth the fecond; the fecond multiplied by 2, produceth the third, and fo on in infinitum. Or in thefe Numbers,

3, 9, 27, 81, 243, &c. where the Common Multiplier is [3.] And this is called Geometrical Progreffion.

Of both these kinds of Progreffion I have inferted Tables.

16 32 48

64

80

96 112 128

144 160

90

108

126144

51 68

17 34
18 36 54

19

20

21

72

57 76

4060

85 102 119 136 153 170

126 | 144 162 180 95 114 133 152 171 190 80 100 120 140 160 180 200 84 105 126 147 168 189 210 88 110 132 1.54 176 198 220 92 115 138 161 184 207 230 96 120 144 168| 192 | 216 | 240 25 50 75 100 125 150 | 175 | 200 | 225 | 250

4263 22 44 66 4466 234669

24 4872

In this Progreffion it is vifible, how Addition and Subtraction in Arithmetical Progreffion, answers to Multiplication and Division in Geometrical Progreffion. For, As in Geometrical Progreffion, 1000 multiplied by 100,000, produces 100, 000, 000: So in Arithmetical Progreffion, the numbers answering to 1000, and 100,000, are (3) and (5,) which being added together make (8,) which anfwers to this Product 100,000,000.

And again, As in Geometrical Progreffion (100, 000, 000) being divided by (100,000) the Quotient is (1000.) So in Arithmetical Progreffion, If from the correfpondent number, belonging to the former Product 100,000,000, (viz. 8,) you fubtract any of the other correfpondent numbers, viz. (5,) the remainder will be the other correfpondent number (viz. 3,) which answers to 1000.

III. Of Mufical Proportion.

Mufical Proportion or Habitude,] is, when the firft number hath the fame proportion to the third, which the difference between the first and fecond hath to the difference between the second and the third.

As in these numbers [3, 4, 6,] whereas 3 is the half of 6: So 1, which is the difference between 3 and 4 the half of 2.

CHA P. IV.

Numerical Theorems.

I. If Numbers (how many foever) exceed one another by an equal Interval ; the Interval between the Greatest and the Leaft, is Multiplex of that equal Interval: according to the multitude of the Numbers propounded, lefs by One.

L'

ET the numbers proposed be these four 1, 3, 5, 7, whose common Interval is 2. Then (by the Hypothefis) 6 is the Interval between the Greatest 7, and the Leaft 1. ---- And likewife the three numbers 2,2,2, are every of them equal to the common difference, and equal one to the other. And the multitude of them (viz. three) equal to the multitude of the number given, (viz. four) lefs by One. And laftly, The Aggregate of these three numbers (viz. fix) is equal to the Interval between 7 the Greatest, and I the Leaft, viz. 6.

Ι

II. If Numbers (how many foever) contain the one the other by an equal Ratio, then the Greatest of thofe Numbers, is Multiplex of the Powers of the Denomination of that equal Ratio multiplied by the Leaft; according to the Multitude of the Numbers given, lefs by One.

L

ET the Numbers given be these four, viz.2,6, 18, 54. And let the Denomination of the Ratio be 3. Then (by the Hypothefis) the first multiplied by 3 (the equal Ratio) is equal to the fecond (viz. 6,) and the fecond multiplied by (3) is equal to the third (viz. 18) and the third multiplied by (3) is equal to the fourth (viz. 54.) Et fic, &c.

C

The

Or, let the four Numbers be 1, 3, 5, 7, and the common difference 2.

Here the Greatest Term (7) is equal to the Leaft (1,) and as many differences as there are more Terms befides the Leaft (viz. fix.) And therefore the Greatest Term 7, lefs by the Least 1, (viz. 6) is Multiplex of the Difference, according to the number of Terms lefs by One.

III. If there be three Numbers in Arithmetical Proportion; the Sum of the two Extreams is equal to the double of the Means.

L

ET the three Numbers be 2, 4, 6, whose common difference is 2.
Then,

The firft Term 2, is only 2.

The fecond Term is 2, more by the difference once (viz. 4.)

The third Term is 2, more by the difference twice (viz 6.) And so it is evident, that twice (2) the firft Term, more by twice the difference (4) is equal both to the Sum of the first and third Terms; and alfo to the double of the mean term (4,) for 2 and 6 is 8; and fo alfo is the double of 4.

IV. If three Numbers be in Proportion, the Number contained under the Extreams, is equal to the Square made of the Mean: And if the Number contained under the Extreams, be equal to the Square of the Mean, thofe three Numbers fhall be in proportion.

L

ET the three Numbers be (2, 6, 18.)

And let the common Ratio be (3.)

The firft Term is (2.)

The fecond Term is 2 into 3 (viz. 6.)

The third Term is 2 into the Square of 3 (viz. 9) equal to 18. And from hence it is evident, That the firft Term drawn into the third (viz. 2 into 18,) is equal to 2 into 2 (viz. 4,) and that into the Square of 3 the common Ratio (viz. 9) For 2 into 18 is equal to (36.) And fo is 4 the fquare of 2, into 9 the fquare of 3 equal to 36. Therefore, the Product of the first multiplied by the third, is equal to the fquare of the Mean.