Mathesis enucleata, or, The elements of the mathematicks by J. Christ. Sturmius ; made English by J.R. and R.S.S.

DEFINITION XXXIV.

HEre remain two things to be taken notice of; First th•• If any whole (quantiy) be so divided into two equ•••• parts(α) 1.1 that the whole, the greater part an•• the less are in a continual proportion; th•• (whole) is said to be cut in extreme and me•• Reason. 2. In a continual Series of that kind 〈◊〉〈◊〉 Proportionals (e. g. 2. 4. 8. 16. 32, &c. or a, 〈◊〉〈◊〉 e{powerof2}a, e{powerof3}a, e{powerof4}a, &c.) the Reason of the first Ter•• to the third(β) 1.2 (2 to 8, or a to e{powerof2}a) is pa••¦ticularly called Duplicate, and to the 4th (〈◊〉〈◊〉 or e{powerof3}a) Triplicate, &c. of that Reason which the same first Te•• has to its second, or any other antecedent of that Series to 〈◊〉〈◊〉 Consequent: But generally these Duplicate and Triplicate Re••¦sons, &c. as others also of the first Term to the third or four•••• of Proportions continually cohering together, (whether the are the same as in the foregoing Examples, or different as 〈◊〉〈◊〉 these, 2, 4, 6, 18, or a, ea, eia, eioa, &c. viz. if the nam•• of the first Reason be e, of the second i, 〈◊〉〈◊〉 the third o, &c.) I say, the Reasons of the fir•••• Term (2 or a) to the third (6 or eia) 〈◊〉〈◊〉 to the 4th (18 or eioa) are said to be compoun••¦ed of the continual intermediate Reasons.

Now from our general Example, what Eucl•• says, is manifest,

CONSECTARY I.

THat the denomination of a compounded Reason arises from the Multiplication of the denominations of the given Simple(α) 1.3 Reasons; as the denomination of the reason com∣pounded of both (viz. a to eia) is produced by multiplying the denomination of the first Reason e by the denomination of the second Reason i, and the denomination of the Reason com∣pounded of the three (viz. a to eioa) is produced by the deno∣mination of the first Reason e, multiplied by the denomina∣tion of the second Reason i; and the Product of these by the denomination of the third Reason o, &c.

CONSECTARY II.

SO that it is very easie after this way, having never so many Reasons given, whether continued (as 2 to 3, 3 to 6, or ••a, ea, eia,) or interrupted or discrete (as 2 to 3, and 5 to 10, or a to ea, and b to i b) to express their compounded Reason: ••n the first case it easily obtain'd by the bare omission of the in∣termediate Term or Terms (2 to 6, or a to eia;) and in the other by multiplying first of all the Names of the com∣pounding Reasons among themselves (1 ½ and 2, e. and i.) and by the Product (3 or ei) as the name of the Reason compound∣ing the first Term (2 or a) that you may have the o her 6 or eia) or (if any one had rather do so in this latter case) by turning the discrete or interrupted Reasons into continued ones, by making as 5 to 10 in the second Reason, so is the Consequent of the first 3 to 6, or as b to ib, so ea to eia,) and then by re∣ferring the first 2 to the third 6, or the first a to the third eia, &c. In a word therefore, any Duplicate Reason may be appositely expressed by a to e{powerof2}a, and Triplicate by a to e{powerof3}a, the one immediately discernible by a double, the other by a triple Multiplication into itself; as you may also commodiously, and denote others compounded, e. g. of 2 by a to eia, of 3 by a to eioa, &c.

SCHOLIUM.

WE will here advertise the Reader, that tho the Names 〈◊〉〈◊〉 duplicate & triplicate Reasons, &c. are chiefly appropriate to Geometrical Proportionality, yet the Moderns have also accom∣modated them to Arithmetical also; as e. g. That Arithmetic•• Progression is called Duplicate, whose Terms are the Squares 〈◊〉〈◊〉 Numbers Arithmetically Proportional (e. g. 1, 4, 9, 16, 25 &c.) and Triplicate, whose Terms are Cubes, (&c. as 1, •• 27, 64, &c.

Notes

• (α) 1.1

  Eucl. De∣finit. 3. lib. 6
• (β) 1.2

  Eucl. De∣finit. 10. l. 5.
• (α) 1.3

  Eucl. l. 6. Def. 5.