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XIV. Some Remarks upon the Nine Digits, as they are difpofed in the Table.

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I. Of the Digit 1.

Ittle can be faid concerning this Digit, for that it neither Multiplieth nor Divideth, but leaveth the Number which is Multiplied or Divided ftill the fame: But this you may take notice of, That any two Terms, taken equidiftant from the Middle Term, do make 10, and the Middle Term (which is 5) doubled does make 10 alfo. ΙΟ

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Again; If you Multiply the laft term (9) by the middlemoft term (5), that Product fhall be equal to the Sum of all the Products added together; for 1, 2, 3, 4, 5, 6, 7, 8, and 9, added together, do make 45. Alfo, If you leave out the last term (9), and take the two extreams fucceffively, and add them together, every of their Sums shall be equal to 9.

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In the Addition of the Products of the Multiplication of this Digit, you have one of every of the Nine Digits, tho not orderly difpofed, yet they ftand fo, that if you add the two extreme terms equidiftant from the middle term together, the Sum of each of them shall be equal to (11).

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And if

you leave out the last term 9, then every two extream terms taken together, shall be equal to (9) the laft term.

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And again, If you add the numbers in the fourth row under this Digit together, you will find the Sum of them to be 45, which multiplied

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by

by the Digit (2) makes 90, equal to the Sum of the Products of all the Multiplications of this Digit, which ftand in the third Row under Digit II.

III. Of the Digit 3.

In this Digit you may obferve (as in all the reft) that the Products of the feveral Multiplications do proceed in an Arithmetical Progreffion, the Common Difference being that of the Digit: So in this of the Digit (3). Wherefore if you add any two terms equidiftant from the middle term, the Sum of those two terms fhall be equal, and the double of the middle term equal thereunto alfo: So in this

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In this Digit you may obferve, that all the Additions of the Products are one of these three Digits, viz. 3, 6, or 9; and now if you multiply every one of thefe feverally by (3) the common difference, the Products of each of them will be 9, 18, 27; which added together do make 54, which is equal to the Sum of the fourth Row under this Digit (3). And in this fourth Row of Additions, if you add the two extream terms, as 3 and 9 together, their Sum will be 12, to which the middle term (6) doubled," is alfo equal.

IV. Of the Digit 4.

In the fourth Row under this Digit (4) you have all the 9 Digits produced; and if you omit the laft term, every two of the reft added together fhall be equal to 9; and the Sum of all that Row added together makes 45; which if you multiply by the digit (4), the Product will be (180) equal to the third Row of all the Products added together.

V. Of the Digit 5.

This Digit hath all the Nine Digits in the fourth Row of Additions (as the First, Second, and Fourth had), and therefore hath the fame qualifications; for omitting the last term, every two figures taken equidistant from the extreams, fhall be equal to (9), and the Sum of all that Row fhall be (45); which multiplied by the common difference (5) the Product fhall be (225) equal to the Sum of all the Products of the Multiplications in the third Row. In which Row alfo, if you add the two extream terms together (5 and 45),the Sum of them is(50), and is equal to the double of the middle term (25) And foalfo in the fourth Row, if you add the two extream Digits (9 and 5) together, their Sum (14) fhall be equal to the double of the middle term (7).

VI. Of the Digit 6.

This Digit hath the fame qualifications as had the Digit (3), for all the fourth Row of Additions are one of these Digits 6, 3, or 9: And omitting the laft term, every two terms taken equidiftant from the extreams, fhall be equal to 9, and the fum of all of them equal to 54, equal to the laft number of the third row: And in the third row, if you add any two numbers equidiftant from the two extreams, their fum will be 60, equal to the double of the middle term (30). And farther; the fum of the fourth row being (54), that multiplied by (5), a number less by one than (6) the common difference, (because it exceeds 5), the Product will be 270, equal to the fum of all the Products in the third row.

VII. Of the Digit 7.

In this Digit (as in all the reft) the third row, which proceeds by multiplying of the Digit 7 into all the other Digits, defcends in Arithmetical Progreffion, the common difference being (7); and fo if you add the two extream numbers together, as (7) and (63), the fum of them will be (70), which will be equal to the double of the middle number (35). And the like will be by the addition of any two of them equidiftant from the two extreams, or from the middle number,

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And fo is the double of ( 35 the middle term alfo) equal to 70.

Again; the laft number in the third being (63) multiply it by (5), a number lefs by 2 than the common difference, and the Product will be (315) equal to the fum of all the Products in that third row added together.

VIII. Of the Digit 8.

In this the third Row increases by an Arithmetical Progreffion, the common difference being (8); fo that the two extream terms (8 and 72) being added together, do make (80), equal to the double of the middle term (40), and fo any two that are equidiftant from the middle term: And for the fum of them, multiply the laft term (72) by (5) and the Product will be (360) equal to the fum of all the Products in the third row: And for the fourth row, they all defcend till they come to (o) the last being (9) as it is in all the other Unites.

IX. Of the Digit 9.

The third row under this Digit proceeds in an Arithmetical Progreffion, fo that the two extream terms (8 and 81) make (90), equal to the double of (45) the middle term; and fo do any two of the reft that are taken equidiftant from both the extreams, or from the middle

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term (45): And for the fum of the numbers in this third row, the laft number (81) multiplied by (5) gives (405) for the fum of all that row. As for the numbers (or Sums) in the fourth row, they are all equal one to the other, namely (9), and they being all added together, their fum is equal to the laft term in the third row.

СНАР. II.

Of Comparative Arithmetick; or of the Relation of Numbers in Quantity.

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Omparative Arithmetick is performed by Numbers, as they are confidered to have relation one to another in Quantity or Quality.

1. Relation in Quantity] is the reference or respect that the Numbers themselves have one to another: As when the comparison is made between 6 and 2, or 2 and 6; 5 and 3, or 3 and 5. And here the numbers propounded are always two, whereof the firft is called the Antecedent, and the fecond the Confequent; fo in the firft Example 6 is the Antecedent, and 2 the Confequent; and in the second, 2 is the Antecedent, and 6 the Confequent.

2. Relation in Quality ], confifts either in the Difference, or elfe in the Rate or Reason that is found between the Terms propounded.

3. Difference.] The Difference of two numbers is the remainder which is left after Subtraction of the leffer out of the greater; fo 6 and 2 being the Terms propounded, 4 is the Difference between them, for 2 fubtracted from 6, the remainder is 4.

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4. Rate or Reafon. ] The Rate or Reafon between two numbers, is the Quotient of the Antecedent divided by the Confequent: So if it be demanded what Rate or Reafon 6 hath to 2; the Answer is, Triple Rea fon; for if you divide the Antecedent 6, by the Confequent 2, the Quotient will be 3, 2 being contained in 6 juft three times. like manner is there fub-triple Reafon between 2 and 6; for if you divide 2 by 6, the Quotient is, or, because 6 being not found once in 2, there remains 2 for Numerator, 6 the Divifor being the Denominator. This Rate or Reafon of Numbers is either Equal, or Unequal.

5. Equal Reafon] Is the relation that equal numbers have one to another, as 5 to 5,- 6 to 6,—————————7 to 7, &c. For here, one being divided by the other, the Quotient is always an Unite; for if it be demanded, how often 5 may be had in 5, the answer will be, 1, or Unity.

6. Unequal Reafon ] is the relation that unequal numbers have one unto another; and this is either of the Greater to the Leffer, or of the Lefs to the Greater. — -Unequal Reafon of the Greater to the Leffer ], is, when the greater term is Antecedent, as of 6 to 2, or 9 to 7, and the like; for here the Quotient of the Antecedent divided by the Confequent, is always greater than Unity; fo 6 divided by 2, the Quotient is 3, and 9 divided by 7, is 13. But Unequal Reafon of the Leffer to the Greater ], is, when the leffer term is Antecedent, as of 2 to 6, or 7 to 9, &c. And here the Quotient of the Antecedeut divided by

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the Confequent, is always lefs than Unity; fo 2 divided by 6, the Quotient is, or, and 7 divided by 9, is

Each of these kinds of Unequal Reafon is again fubdivided into Five other kinds or varieties; whereof the three firft are Simple, and the other two are Mixt. -The Simple kinds of Unequal Reason are, (1.) Manifold. (2.) Superparticular. (3.) Superpartient.

7. Manifold Reafon of the Greater to the Lefs], is when the Confequent is contained in the Antecedent divers times, without any part remaining; as 4 to 2, 8 to 4, 16 to 8; which is called Double Reafon, because the Lefs is contained twice in the Greater; fo 6 to 2 is Triple Reafon, and 8 to 2 Fourfold Reafon, &c. And here, the Quotient of the Antecedent divided by the Confequent, is always a whole number 8 divided by 2, the Quotient is a whole number. -The opposite of this kind, viz. Of the Lefs to the Greater, is called Sub-manifold: Examples hereof are, 2 to 4, 4 to 8, 8 to 16, &c. Also 2 to 6, 2 to 8, 2 to 10, &c.

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8: Superparticular Reafon of the Greater to the Leffer ], is, when the Antecedent contains the Confequent once; and befides, an Aliquot part of the Confequent; that is, one half, one third, one fourth, or one fifth, &c. of the Confequent; as 3 to 2, 4 to 3, 5 to 4, 6 to 5, and the like: So here, 3 divided by 2, the Quotient is 1 and 4 divided by 3, the Quotient is I; 5 by 4 the Quotient is 1; wherefore I fay, 2 and half 2 (which is 1) do conftitute 3; fo3, and of 3 do conftitute 4; and fo of the rest. For here, the Quotient of the Antecedent divided by the Confequent, is a Mixt Number, whose whole part, and the Numerator of the Fraction, is always an Unite. -The oppofite Reafon of this kind is Subfuperparticular, as 2 to 3, 3 to 4, 4 to 5, 5 to 6, &c. 9. Superpartient of the Greater to the Lefs ], is, when the Antecedent contains the Confequent once, and divers parts of the Confequent befides; as 5 to 3, 7 to 5, 7 to 4, 8 to 5, 9 to 5, 11 to 7, &c. Here 5 divided by 3, the Quotient is 1, and therefore 5 contains 3 once, and } of 3, which is 2, and they two together do conftitute 5. the Quotient of the Antecedent divided by the Confequent, is a Mixt Number, whose whole part being a Unite, hath always for the Numerator of the Fraction a Number compofed of more Unites than one; fo the conference being made between 5 and 3, and 5 the Antecedent being divided by 3 the Confequent, the Quotient is 13 -The oppofite of this Reafon is Subfuperpartient; Examples hereof are, 3 to 5, 5 to 7, 4to 7, 5 to 8, 5 to 9, 7 to 11, &c.

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The Mixt kinds of Unequal Reafon are two, viz. (1.) Manifold Superparticular. And (2.) Manifold Superpartient.

10. Manifold Superparticular Reafon ], is, when the Antecedent contains the Confequent divers times, and an Aliquot part of the Confequent befides; as 5 to 2, 10 to 3, 17 to 4, 21 to 5, and the like. Here the Quotient of the Antecedent divided by the Confequent, is a Mixt Number, whose whole part confifting of more Unites than one, hath always an Unite for the Numerator of the Fraction annexed unto it; fo 5 di. vided by 2, hath for the Quotient 2, and 21 divided by 5, the Quoti

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