The compleat arithmetician, or, The whole art of arithmetick, vulgar and decimal in a plain and easie method, suitable to the meanest capacity : in which the multiplication and division of numbers of several denominations, and the rule of alligation are more fully explained than in any treatise of this nature, yet extant / by J.N., Philomath.

CHAP. XX. Of the relation of numbers in Quality.

1. RElation of numbers in Quality, (other∣wise called Proportion,) is either A∣rithmetical or Geometrical.

2. Arithmetical proportion, is a Relation that numbers have unto the equality of their differen∣ces; as in 8. 7. 6. and in 12. 9. 6. 3 there are two proportions which have equal differences; in that first rank of numbers, the equal or common dif∣ference is 1, and in the second 3.

3. Arithmetical proportion, is either continu∣ed or interrupted.

4. Arithmetical proportion continued, is when the same difference is continued from the

first numbers to the last, as in this rank of num∣bers. 5. 7. 9. 11. the common difference is 2.

5. Arithmetical proportion interrupted, is when the difference between the first and second is the same with the third and fourth, but not between the second and third, or when the difference is discontinued in any part of the rank, as in these numbers, 10. 8. 4. 2. the difference between 10 and 8 is 2, and the difference between 2 and 4 is 2, but the difference between 4 and 8 is 4.

6. In Arithmetical progression, two things may be inquired, either the terms that constitute the pro∣gression, or the sum that the terms in a certain pro∣gression made.

7. Sometimes all the terms in an Arithmetical progression at a certain given rate may be requi∣red to be expressed in their natural order; or the progression being interrupted, some particu∣lar term in that progression may be required.

8. If it be required to express the terms in a∣ny rate propounded, add the rate to any given term continually and you have what is required; as if it were required to make a rank of numbers, in Arithmetical proportion, answerable to the rate or difference that is between 1 and 5, add 4 which is the difference between the numbers gi∣ven unto 5, and so continually to their sum and you shall constitute this rank of numbers 1. 5. 9. 13. 17. 21. which is the rank of numbers in Arith∣metical proportion required.

9. If some particular term in any progression be required without the rest, deduct, from the term required, the remainer being multiplied by

the common difference in that progression, and the product added to the first shall give the term desired: as in this progression 4. 7. 10, let the 12 term from 4 the first term given be required, de∣ducting one from 12 the remainer is 11, which being multiplied by 3 the common difference the product is 33, to which adding 4 the first term, the sum is 37 for the 12 term required: and in like manner the 30 term wil be found to be 91: the like may be done for any other.

The Terms.	4	2	10	37	91
The number.	1	2	3	12	30

10. If the sum of the terms in any rank of numbers, in Arithmetical proportion continued, multiply the sum of the first and last terms by the number of the terms, half the product shall be the sum required, as in this example, 1. 5. 9. 13. 17. 21, the sum of the first and last terms is 22, which being multiplied by 6 the number of the terms the product is 132, the half whereof is 66 the sum of the terms required.

11. Three numbers being given which differ by Arithmetical proportion continued, the mean number being doubled, shall be equal to the sum of the extreams, so 5. 9. 13. being given, the double of 9 is 18 and the sum of 5 and 13 is 18 also.

12. Four numbers being given that differ by Arithmetical proportion continued or interrup∣ted, the sum of the two means shall be equal to

the sum of the two extreams, so 5. 9. 13 17 be∣ing given, the sum of 9 and 13 the two mean numbers, is equal to the sum of 5 and 17 the 2 extreams.

13. Geometrical proportion is a relation that numbers have to the equality of their rate or rea∣son: as in 2. 6. 4. 12, where the rate between 2 and 6 is the same with that which is between 4 and 12; for as 6 is three times as much as 2, so 12 is three times as much as 4.

14. Geometrical proportion is either conti∣nued or interrupted.

15. Geometrical proportion continued, is when the same rate or reason is still kept in the whole rank of numbers given, as 2. 4. 8. 16. 32, in which the progression is continued by double rea∣son, for as 4 is twice 2, so 8 is twice 4, 16 is twice 8, and 32 is twice 16.

16. In numbers that increase by Geometrical proportion continued, if you multiply the last term by the rate or reason by which the rank of numbers is created, and from the product sub∣tract the first, the remainer being divided, by the rate less one shall give you in the quotient, the total sum of all the terms. Example, let the rank of numbers propounded be 3. 9. 27. 81. 243, the rate or reason by which this rank of num∣bers is created is 3, by which the last term 243 being multiplied the product is 729, out of which deducting 3 the first term, the remainer is 726, which being divided by the triple less one that is by 2, the quotient gives me 363, for the sum of the terms propounded.

17. Three numbers being given which do in∣crease by Geometrical proportion, the square of the mean shall be equal to the product of the extreams; as 9. 27. 81. being propounded, the square of 27 is 729, and the product of 81 by 9 the two extreams is 729 also.

18. Geometrical proportion interrupted, is when the rate or reason which is between the first and second and between the third and fourth, is not to be found between the second and third; as 6. 3. 16. 8. in which 3 is half 6, and 8 half of 16, but the same rate is not found between 3 and 16.

19. Four numbers being given which do in∣crease by Geometrical progression continued or interrupted, the product of the two mean num∣bers shall be equal to the product of the extreams: let the four numbers given be 6. 3. 16. 8. the pro∣duct of 16 by 3 is 48, and the product of 8 by 6 the extream numbers is 48 also.