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

Proposition XX.

IF there are never so many Continual Proportionals, the Pro∣duct of the Extremes is equal to the Product of any 2 of the Means that are equally distant from the Extremes, as also to the Square of the mean or middle Term, if the Terms are odd.

Demonstration.

Such are e. g. a, ea, e{powerof2}a, e{powerof3}a, e{powerof4}a, e{powerof5}a, e{powerof6}a, &c. and the Product of the Extremes, and of any two Terms equally remote

them, and the Square of the mean or middle Term, every where e{powerof6}aa. Q. E. D.

SCHOLIUM I.

NOR can there be any doubt but this will always be so, how far soever the Progression is continued; if 〈◊〉〈◊〉 con∣sider that the last Term always contains the first, 〈◊〉〈◊〉 way of Reason, so many times multiplied as is the place of that Term in the rank of Terms, excepting one. Altho therefore the last Term but one is in one degree of its Reason less than 〈◊〉〈◊〉 last, ••he second on the contrary, is in one more than the 〈…〉〈…〉 re∣••ore the Product of the one will necessarily 〈…〉〈…〉 ••e Product of the other. Thus also the las•• 〈…〉〈…〉 Degrees of Proportion lower than the 〈…〉〈…〉 being to be multiplied into that, exc••••ds 〈◊〉〈◊〉 fi st 〈…〉〈…〉 of the Proportion, &c. as may be seen thou U••iver••al Ex•••••••••• Hence you have the following

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).

SCHOLIUM II.

THat we may exhibit the whole Reason of this admirabl•• Artifice (which about 35 years ago was found out b•• the Honourable Lord John Naper Baron of Merchiston in Scotland and published something difficult, but afterwards render'd much easier and brought to perfection by Henry Briggs, the first S••¦vilian Professor of Geometry at Oxford.) I say that we may exhibit the whole Reason of it in a Synopsis, after an easie way when its use appear'd so very Considerable in the great Num∣bers in the Tables of Sines and Tangents, nor yet could they be useful without mixing vulgar Numbers with them, especial∣ly in the Practical Parts of Geometry, the business was to ac∣••ommodate

this Logarithmical Artifice to them both. First ••herefore that Artists might assign Logarithms to all the com∣mon Numbers proceeding from 1 to 1000 and 10000, &c. ••hey first of all pick out those which proceed in continued Geo∣metrical Proportion, and particularly, tho arbitrariously, those which increase in a Decuple Proportion, e. g. 1. 10. 100. 1000. 10000, &c.

But now to fit them according to the Foundation of Consect. 8. a Series of Ordinals in Arithmetical Progression, we do'nt only substitute the simple Number 1, 2, 3, &c. but augmented with several Cyphers after them, that so we may also assign ••heir Logarithms in whole Numbers to the intermediate Num∣••ers between 1 and 10, 10 and 100, &c. Wherefore, by ••his first Supposition, Logarithms in Arithmetical Proportion, ••nswer to those Numbers in Geometrical Proportion, after the ••ay we here see,

1	10	100
Log. 0000000	10000000	20000000
	1000	10000
	30000000	40000000, &c.

As that they also exhibit certain Characteristical initial Notes, whereby you may see, that all the Logarithms between 1 and ••0 begin from 0, the rest between 10 and 100 from 1, the ••ext from 100 to 1000 from 2, &c.

The Logarithms of the Primary Proportional Numbers being ••hus found, there remain'd the Logarithms of the intermediate Numbers between these to be found: For the making of which, ••fter different ways, several Rules might be given drawn from ••he Nature of Logarithms, and already shewn in Consect. 3. See Briggs's Arithmetica Logarithmica, and Gellibrand's Trigonometria Britannica; the first whereof, chap. 5. and the following, shews ••t length both ways delivered by Neper in his Appendix. But ••he business is done more simply by A. Ʋlacq. in his Tables of Sines &c. whose mind we will yet further explain thus: If you are ••o find, e. g. the Logarithm of the Number 9, between 1 and ••0, augmented by as many Cyphers as you added to the Lo∣garithm of 10, or the rest of the Proportionals (h. e. between

10000000 and 100000000) you must find a Geometric•• Mean Proportional, viz. by multiplying these Numbers togeth•••• and extracting the Square Root out of the Product, by Pr•• 17. Now if this Mean Proportional be less than 9 augmen•••• by as many Cyphers, between it and the former Denary Nu••∣ber you must find a second mean Proportional, then betwe•••• this and that same a third; and so a fourth, &c. but if it 〈◊〉〈◊〉 greater, then you must find a mean Proportional between and the next less, &c. till at length after several Operatio•• you obtain the number 9999998, approaching near 90000000. Now if between the Logarithm of Unity a•• Ten (i. e. between 0 and 10000000) you take an Arithm••¦tical Mean Proportional (05000000) by Bisecting their S•• by Prop. 14. and then between this and the same Logarithm Ten, you take another mean, and so a third and a fourth, 〈◊〉〈◊〉 at length you will obtain that which answers to the last abo•• mentioned, viz. 9. See the following Specimen.

A TABLE of the Geometrical Proportionals betwe•••• 1 and 10, augmented by 7 Cyphers, and of t•••• Arithmetical Proportionals between 0 and 10000000 being the Logarithms corresponding to them.

Geometrical Mean Pro∣portionals.		Arithmetical Logar. mean Proportionals
31622777	First,	05000000
56234132	Second,	07500000
74989426	Third,	08750000
86596435	Fourth,	09375000
93057205	Fifth,	09687500
89768698	Sixth,	09531250
91398327	Seventh,	39609375
90579847	Eighth,	09570312
90173360	Ninth,	09550781
89970801	Tenth.	09541015

Which is thus made: In the first Table a Geometrical Mean ••••oportional between 10 000 000 and 100 000 000 the first Number of it; then another Mean between that and ••e same last 100000000, gives the second; and so to the ••••th, 93057205. Which, since it is already greater than the ••ovenary, another Mean between it and the precedent fourth, ••••comes in order a sixth, but sensibly less than the Novenary. ••herefore between it and the fifth you will have a seventh ••ean yet greater than the Novenary; and between the sixth ••••d seventh, an eighth, somwhat nearer to the Novenary, but ••t yet sensibly equal, but somewhat bigger; moreover between ••••e sixth and eighth you will have a ninth, between the ninth ••••d sixth a tenth gradually approaching nearer the Novenary, but ••••t somewhat sensibly differing from it. Now if you con∣••ue this inquiry of a mean Proportional between this tenth, 〈◊〉〈◊〉 somewhat too little, and the precedent ninth as somewhat ••o big, and so onwards, you will at length obtain the Num∣••••r 8999 9998, only differing two in the last place from the ••ovenary Number augmented by seven Cyphers, and conse∣••ently insensibly from the Novenary it self. But for the Lo∣••arithm of this in the second Column, by the same process you ••••e to find Arithmetical Mean Proportionals between every 2 ••ogarithms answering to every two of the superiour ones, till you ••nd, e. g. the Logarithm of the tenth Number 09541015, ••d so at length the Logarithm of the last, not sensibly differ∣••••g from the Novenary, 09542425.

Thus having found, with a great deal of labour, but also ••ith a great deal of advantage to those that make use of them, ••••e Logarithms of some of the numbers between 1 and 10, and ••0 and 100, &c. you may find innumerable ones of the other ••ntermediate Numbers with much less labour, viz. by the help ••f some Rules, which may be thus obtain'd from Consect. 3 of ••e precedent Proposition. The Sum of the Logarithms of the ••umber Multiplying and the Multiplicand, gives the Logarithm of the ••roduct. 2. The Logarithm of the Divisor subtracted from the Lo∣••arithm of the Dividend, leaves the Logarithm of the Quotient: ••he Logarithm of any number doubled, is the Logarithm of the Square, ••ripled of the Cube, &c. 4 The half Logarithm of any number is ••he Logarithm of the Square Root of that number, the third part of 〈◊〉〈◊〉 the Cube Root, &c. Thus, e. g. if you have found the Lo∣garithm

of the number 9, after the way we have shewn, by th•• same reason you may find the Logarithm of the number 5 (vi•• by finding mean Proportionals between the second and the fi•• number of our Table, and between their Logarithms, &c. and by means of these 2 Logarithms you may obtain several o¦thers: First, since 10 divided by 5 gives 2; if the Logarith•• of 5 be subtracted from the Logarithm of 10, you'l have th•• Logarithm of 2, by Rule the second. Secondly, since 10 m••••¦tiplied by 2 makes 20, and by 9 makes 90, by adding th•• Logarithms of 10 and 2, and 10 and 9, you'l have the L••¦garithms of the numbers 90 and 20, by Rule 1. Thirdly Since 9 is a Square, and its Root 3, half the Logarithm of 〈◊〉〈◊〉 gives the Logarithm of 3, by Rule 4. since 90 divided by 〈◊〉〈◊〉 gives 30, the Logarithm of this number may be had by s••••¦tracting the Logarithm of 3 from the Logarithm of 90, b•• Rule the second. Fifthly, 5 and 9 squared make 25 and 8▪ the Logarithms of 5 and 9 doubled, give the Logarithms 〈◊〉〈◊〉 these numbers, by Rule 3. In like manner, sixthly, the Su•• of the Logarithms of 2 and 3, or the Difference of the L••¦garithms of 5 and 30, give the Logarithm of 6, and the Su•• of the Logarithms of 3 and 6, or 2 and 9, gives the Log••¦rithm of 18; the Logarithm of 6 doubled, gives the Loga¦rithm of 36, &c. And after this way you may find and reduce it to Tables, the Logarithms of Vulgar Numbers from 1 to 100•• (as in the Tables of Strauch. p. 182, and the following) or 〈◊〉〈◊〉 100000 (as in the Chiliads of Briggs) But as to the manner ••¦deducing the Tables of Sines and Tangents from these Loga¦rithms of Vulgar Numbers, we will shew it in Schol. of Pr•••• 55, only hinting this one thing before-hand; that this Artifi•• of making Logarithms is elegantly set forth by Pardies in hi•• Elements of Geometry, pt 112. by a certain Curve Line then•••• called the Logarithmical Line; by the help whereof he suppose Logarithms may be easily made; and having found those o•• the numbers between 1000 and 10000, he shews, that all o∣thers may be easily had between 1 and 1000. Wherefore w•• shall Discourse more largely in Schol. Definit. 15. lib. 2.