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

CONSECTARIES.

••. HAving some of the Terms given in a Continual Pro∣portion (e. g. suppose 10) you may easily find any other that shall be required (e. g. the 17th) as the last; If ••he 2 Terms given, being equally remote from the first and ••hat required (as are e. g. the eigth and tenth) be multiplied by one another, and this Product, like that also of the Extremes, be divided by the first.

II. But this may be performed easier, if you moreover take ••n this Observation, That if, e. g. never so many places of pro∣portionals, passing over the the first, be noted or marked by Ordinals or Numbers according to their places (as in this uni∣versal Example)

a,	ea,	e{powerof2}a,	e{powerof3}a,	e{powerof4}a,	e{powerof5}a,	e{powerof6}a,
	I.	II.	III.	IV.	V.	VI.

The place of the 7th Term is (e. g.) VI. (and so the place of any other of them being less by Unity than its number is among the Terms) and also composed of the places of any other equal∣••y distant from the Extremes, e. g. V. and I. IV. and II. or ••wice III. &c.

III. Here you have the Foundation of the Logarithms, i. •• of a Compendious Way of Arithmetick, never enough to b•• praised. For if, e. g. a rank of Numbers from Unity, con••••∣nually Proportional, be signed or noted with their Ordinals, as w•• have said, as Logarithms,

1.	2.	4.	8.	16.	32.	64.	128.	256,	&c.
	I.	II.	III.	IV.	V.	VI.	VII.	VIII.

and any two of them (as 8 and 32) are to be multiplied to¦gether; add their Logarithms III and V, and their Sum VII•• gives you the Logarithm of their Product 256, as the Te•••• equally remote from the 2 given ones and the first, and 〈◊〉〈◊〉 whose Product with the first (which is Unity) i. e. it self w•••••• be equal to the Product of the Numbers to be multiplied: A•••• contrariwise, if, e. g. 128 is to be divided by 4, subtracting t•••• Logarithm of the first II from the Logarithm of the second V•• the remaining Logarithm V points out the number sought 3•• so that after this way the Multiplication of Proportionals 〈◊〉〈◊〉 by a wonderful Compendium, turned into Addition, and the Division into Subtraction, and Extraction of the Square Ro•••• into Bisecting or Halving, (for the Logarithm of the Squa•••• Number 16 being Bisected, the half II gives the Root sough•• 4) of the Cube Root into Trisection (for the Logarithm of th•• Cube 64 being Trisected, the third part gives the Cubi•• Root sought 4).