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

CHAP. IV. Of Geometrical Proportion in General.

Proposition XVII.

IF there are three Quantities continually (α) Proportional, the Rectangle of the Extremes, is equal to the Square of the mean Term.

Demonstration.

Such are e. g. a, ea, e{powerof2}a,	The mean Term, 〈 math 〉〈 math 〉
The Extremes
Rectangle	Square Q.E.D.

SCHOLIUM.

MOreover if three Quantities on each side are in the same Continual Proportion, as

[illustration]

the Rectangles of the Extremes made Cross-ways, are equal to the Rectangle of the mean Term; being every way e{powerof2}ab.

Whence by the way may appear that Proposition of Archimedes(α) 1.1 (β) 1.2 That the Surface of a Right Cone is equal to the Circle, whose Radius is a mean Proportional between the Side of that Cone and the Semidiameter of the Base. For suppose EF to be a mean Pro∣portional between the side of the Cone BC (Fig. 57.) and the Semidiameter of the Base CD, since an equal number of Peripherys answer to an equal number of Radii in the same Propor∣tion; half the Product of the first Line BC into the last Periphery, ½ e{powerof2}ab (that is, by Consect. 4. Definit. 18. the Surface of the given Cone) will be equal to half Product of the mean Line into the mean Periphery, ½ e{powerof2}ab (i. e. by Consect. 2. Definit. 15.) to the Area of the Circle of the mean Proportional EF. Q. E. D.

The same Proposition of Archimedes may also be Demonstrat∣ed after this Way: If the side of the Cone BC be called b, and the Semidiameter of the Base a) so that the Periphery may, by Consect. 1. Definit. 31. be 2ea, and so the Surface of the Cone, by Consect. 4. Definit. 18. eab) the √ab will be a mean propor∣tional between b and a, by this 17th Proposition; which being taken for Radius, the whole Diameter will be √2ab, and the Pe∣riphery 2e√ab; therefore by Consect. 2. Definit. 15. half the Ra∣dius ½√ab multiplied by the Periphery (since √ab multiplied, by √ab necessarily produces ab) will give you the Area of the Cir∣cle by that mean(a) 1.3 Proportional, equal to the Surface of the given Cone, which before was expressed in the same Terms. Q. E. D.

Hence also naturally flows this other Proposition, That the Surface of the Cone (½e{powerof2}ab) is to its Base (½ ab) as the Side of the Cone (e{powerof2}b) is to the Radius of the Base b, as may appear from the Terms.

Proposition XVIII.

IF(b) 1.4 4. Quantities are Proportional, either continuedly or dircretely, the Product of the Extremes is equal to the Product of the Means.

Demonstration.

Suppose one Continual Proportional, a, ea, e{powerof2}a, e{powerof3}a.

Extremes e{powerof3}a	Means e{powerof2}a
a	ea
Prod. e{powerof3}aa = Prod. e{powerof3}aa. Q.E.D.

SCHOLIUM.

ON this Theorem is founded the Rule of Three in Arithme∣tick; so called because having 3 Numbers, (2. 5. 8.) it finds an unknown fourth Proportional. For altho this fourth be, as we have said, unknown, yets its Product by 2 is known, because the same with the Product of the Means, 5 and 8. Wherefore the Rule directs to multiply the third by the second, that you may thereby obtain the Product of the Extremes: which divided by one of the Extremes, viz. the first, necessa∣rily gives the other, i. e. the fourth sought.

Proposition XIX.

IF 2 Products (on the other side) arising from the Multipli∣cation of 2 Quantities, are equal, those 4 Quantities will be at least directly Proportional.

Demonstration.

Suppose eba be the equal Product of the Extremes, and eab of the Means; the Extremes will either be eb and a, or e and ba, or b and ea, as also the Means. But what way soever either is taken, there can be no other Disposition or placing of them, than one of the following.

1 eb	eb	a	a
—	ee	ab
—	ea	b;	or inversly.
—	a	eb
—	ab	e

—	b	ea
2 e e	e	ba	ba
—	eb	a
—	ea	b;	or inversly.
—	ba	e
—	a	eb
—	b	ea
3 b	b	ea	ea
—	a	eb
—	ba	e;	or inversly.
—	ea	b
—	eb	a
—	e	ba;	or inverting the Order of them all.

In all these Dispositions there may be immediately seen a Geometrical Proportion, by what we have in Definition 3•• and 33.

CONSECTARSY.

I. AS we have shewn one Sign of Proportionality in the Definition of it, viz. That the same Quotient will a∣rise by dividing the Consequents by the Antecedents; so now we have another Sign of it, viz. The Equality of the Products of the Extremes and Means.

II. By a bare Subsumption may hence appear the Truth of Prop. 14. lib. 6. Euclid. at least partly: Which we shall yet more commodiously shew hereafter.

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.

Proposition XXI.

IF the first Term of never so many Continual Proportionals, be sub∣tracted from the last, and the Remainder divided by the name of the Reason or Proportion lessen'd by Ʋnity, the Quotient will be equal to the Sum of all except the last.

Demonstration.

	ea
	e{powerof2}a
	e{powerof3}a
	e{powerof4}a
	e{powerof5}a
The last Term less the first	e{powerof6}a−a

Divided by the name of the Reason lessen'd by unity.
e−1
* Quote. e{powerof5}a+e{powerof4}a+e{powerof3}a+e{powerof2}a+ea+a; And it is evident from the Operation, that the same will always happen tho the number of Terms be con∣tinued never so far.
e−1
e−1
	e−1
	e−1
	e−1
	e−1

CONSECTARYS.

I. WHerefore in adding never so great a Series of Geo∣metrical Proportionals, since it is enough that the first and last Term, and the Name of the Reason be known, by this Prop. and having found at least some of the Terms of the Proportion, any other may be afterwards found, whose place will be compounded of the places of the two Antecedent ones, according to Consect. 2. Prop. 20. viz. by Multiplying the Terms answering to the two above-mentioned places, and dividing the Product by the first Term; thence it will be very easie to add a great Series of Proportionals into one Sum, tho the particular separate Terms remain almost all of them unknown.

SCHOLIUM.

THese are the same Practical Arithmetical Rules concerning Geometrical Progressions; for the illustration of whic•• Swenterus in Delic. has given us so many pleasant Examples, li•• 1. Prop. 59. and fol. First of all, that famous Example is of th•• kind which relates to the Chequer-work'd Table or Board t•• fling Dice on, with its 64 little Squares, which Dr. Wa•••• has translated out of the Arabick of Ebn Chalecan, into Latin in Oper. Mathem. part. 1. Chap. 31. for the illustration of whic•• we have heretofore composed an Exercitation, and shall he•• only note these few things: If there are supposed 64 Terms 〈◊〉〈◊〉 double Proportion from Unity, and the first of them, note•• with their local Numbers, are these that follow;

1	2	4	8	16	32	64	128
	I	II	III	IV	V	VI	VII

You may have the Term of the 13th place, 8192, by mu••¦tiplying together the VIth and VIIth place; and the Ter•• of the XXVIth place, by squaring or multiplying this new Pro¦duct again by it self, and moreover the Term of the L•••• place, by multiplying that Product again by itself; and furthe•• more the Term of the LIXth place, by multiplication of the num¦ber last found by the number of the VIIth place, and lastly the Term of the LXIIId place (i. e. the last in the proposed Se¦ries) by multiplying this last of all by the number of the IV•••• place.

II. Moreover you may, by this Art, collect infinite Seri•• of Proportional Terms into one Sum, altho it is impossible 〈◊〉〈◊〉 run over all the Terms separately, because infinite. e. g. in 〈◊〉〈◊〉 continued Series of Fractions, decreasing in a double Propo••¦tion ½ ¼, ⅛ 〈 math 〉〈 math 〉, 〈 math 〉〈 math 〉, &c. ad infinitum, if you take them bac••¦wards, you may justly reckon a Cypher or 0, for the fi•••• Term (for between ½ and 0 there may be an infinite Numbe•• of such Terms) and the infinite Sum of these Terms will b•• precisely equal to Unity; for subtracting the first 0, from the last ½, and the remainder ½ being divided by the name of th•• Reason lessened by 1, i. e. by I. which divides nothing, th••

Quotient ½ is the Sum of all the Terms excepting the last, by Prop. 21. and so the last ½ being added, the Sum of all in that Series will be I. Now if the last is not ½ but I, the Sum of all will necessarily be 2; if 2 be the last, the Sum of all will be 4; in a word, it will be always double the last Term.

III. And since in this case the Sum of all the precedent Terms is equal to the last Term, the one being subtracted from the other, there will remain nothing, i. e. ½−¼−⅛−〈 math 〉〈 math 〉−〈 math 〉〈 math 〉, &c. in Infinitum, is = 0, and also 1−½−¼, &c. or 2−1−½−¼. &c. = 0.

IV. In like manner the Sum of infini••e Fractions decreasing in triple Reason in an infinite Series (⅓+〈 math 〉〈 math 〉+〈 math 〉〈 math 〉+〈 math 〉〈 math 〉, &c.) will be equ••l to ½: for if from the last ⅓ (again in an inverted Order) you subtract the first 0, and the Remainder ⅓ be divided by the name of the Reason lessen'd by Unit, i. e. by 2, the Quotient ⅙ will be the Sum of all the antecedent Terms, and adding to this last ⅓ or 〈 math 〉〈 math 〉 the Sum of all will be 〈 math 〉〈 math 〉 or ½.

V. Thus an infinite Series of Fractions decreasing from ¼ in a Quadruple Proportion (¼+〈 math 〉〈 math 〉+〈 math 〉〈 math 〉 &c.) is equal to ⅓; for subtracting the first 0 from the last ¼, and the remainder ¼ being divided by the name of the Proportion, i. e. by 3, you will have 〈 math 〉〈 math 〉 the sum of all except the last, and adding also the last ¼ or 〈 math 〉〈 math 〉, you'l have the whole Sum 〈 math 〉〈 math 〉 or ⅓.

VI. Thus also an infinite Series decreasing from ⅕ in a Quin∣tuple Proportion (⅕+〈 math 〉〈 math 〉+〈 math 〉〈 math 〉, &c.) is equal to ¼: ⅙+〈 math 〉〈 math 〉+〈 math 〉〈 math 〉, &c. is equal to ⅕ &c. and so any Series of this kind is equal to a Fraction, whose Denominator is less by an Unit than the Denominator of the last Fraction in that Series.

VII. Generally also, any infinite Series of Fractions decreasing according to the Proportion of the Denominator of the last Term, and having a common Denominator less by an unit than the Denominator of the last Term (e. g. ⅔+〈 math 〉〈 math 〉+〈 math 〉〈 math 〉, &c. or ¾+〈 math 〉〈 math 〉+〈 math 〉〈 math 〉, &c. or ⅘+〈 math 〉〈 math 〉+〈 math 〉〈 math 〉, &c.) is equal to Unity, after the same way as the Series Consect. 2. which may be compre∣hended under this kind, and which may be demonstrated in all its particular cases by the same method we have hitherto made use of, or also barely subsumed from Consect. 4, 5, and 6. For since ⅓+〈 math 〉〈 math 〉+〈 math 〉〈 math 〉, &c. is equal to ½; ⅔+〈 math 〉〈 math 〉+〈 math 〉〈 math 〉 will be equal to 〈 math 〉〈 math 〉, or 1, and so in the rest.

VIII. Particularly the sum of 〈 math 〉〈 math 〉, &c, decreasing in a Quadruple Proportion, is equal to 〈 math 〉〈 math 〉; and the sum of 〈 math 〉〈 math 〉, &c. is equal to 〈 math 〉〈 math 〉; and the sum of 〈 math 〉〈 math 〉, &c. decreasing in Octuple Proportion, is equal to ⅛: For subtracting the first Term 0, and dividing the remainder by the name of the Reason lessen'd by 1, i. e. by 3, the Quotient 〈 math 〉〈 math 〉 gives the sum of all except the last. This therefore (viz. 〈 math 〉〈 math 〉) being ad∣ded, the sum of all will be 〈 math 〉〈 math 〉 or 〈 math 〉〈 math 〉: In like manner 〈 math 〉〈 math 〉 being divided by the name of the Reason lessen'd by Unity, the Quo∣tient will give 〈 math 〉〈 math 〉, and adding the last, the sum of all will be 〈 math 〉〈 math 〉 i. e. ⅛. So that hence it is evident, that 〈 math 〉〈 math 〉, &c. or −〈 math 〉〈 math 〉, &c. in Infinitum, will be equal to noth∣ing; also ⅛−〈 math 〉〈 math 〉−〈 math 〉〈 math 〉−〈 math 〉〈 math 〉 &c. = 0.

IX. The Sum of a simple Arithmetical Progression (i. e. ascending by the Cardinal Numbers) continued from 1, ad Infinitum, is S••••∣duple of the Sum of the same number of Terms, each of which is equ•••• to the greatest; or on the contrary, this latter Sum is double of the fo••∣mer. We might have subsumed this in Consect. 4. Prop. 16. for, prefixing a Cypher before Unity, it will be a case of that Con∣sectary, the Sum of the Progression remaining still the same. B•••• that this is true, in an infinite Series beginning from Unity (f•••• in a finite or determinate one, the proportion of the Sum is al∣ways less than double, tho it always approaches to it, and come so much the nearer by how much greater the Series is) 〈◊〉〈◊〉 shall now thus Demonstrate: To the Sum of three Terms, 〈◊〉〈◊〉 2, 3, i. e. 6, the sum of as many equal in number to the greatest, i. e. 9, has the same Proportion as 3 to 2; but t•• the sum of six Terms, 1, 2, 3, 4, 5, 6, i. e. 21, the su•• of as many equal to the greatest, i. e. 36, has the same pro∣portion as 3 to 1+¾, that is, as 3 to 2−¼, the decrease be∣ing ¼: but to the sum of 12 Terms, which may be found b•• Consect. 1. Prop. 16.=78, the sum of so many equal to the greatest, viz. 144. has the same proportion (dividing both sid•• by 48) as 3 to 1 〈 math 〉〈 math 〉, i. e. 3 to 1+½+⅛ (for 24 make ½, a•••• the remainder 〈 math 〉〈 math 〉 is the same as ⅛) that is, as 3 to 2−¼−•••• the decrement being now ⅛. Since therefore, by doubling th•• number of Terms onward, you'l find the decrement to be 〈 math 〉〈 math 〉, an•• so onwards in double Proportion; the sum of an infinite Num∣ber of such Terms, in Arithmetical Progression, equal to th•• greatest

will be to the sum of the Progression from 1, ad Infinitum, as 3 to 2−¼−⅛−〈 math 〉〈 math 〉, &c. that is, by Consect. 2 and 3, as 3 to 2−½, that is, as 3 to 1 ½, or as 2 to 1. Q.E.D.

X. The Sum of any Duplicate Arithmetical Progression (i. e. a Progression of Squares of whole numbers ascending) continued from 1 ad Infinitum, is subtriple of the Sum of as many Terms equal to the greatest as is the number of Terms: For any such finite Progression is greater than the subtriple Proportion, but approaches nearer and nearer to it continually, by how much the farther the Series of the Progression is carried on. Thus the Sum of 3 Terms 1, 4, 9=14 is to thrice 9=27 as 1 〈 math 〉〈 math 〉, or 1 〈 math 〉〈 math 〉, or 1+½+〈 math 〉〈 math 〉 to 3 (dividing both sides by 9,) the Sum of six Terms, 1, 4, 9, 16, 25, 36, viz. 91. to six times 36, i. e. to 216 (dividing both sides by 72) is as 1+¼+〈 math 〉〈 math 〉 to 3; and the Sum of 12 Terms 650, to 12 times 144, i. e. to 1728 (dividing both sides by 576) is as 1+⅛+〈 math 〉〈 math 〉 to 3, &c. the Fractions adhering to them thus constantly decrea∣sing, some by their half parts, others by three quarters (for 〈 math 〉〈 math 〉 is 〈 math 〉〈 math 〉; therefore the first decrement is 〈 math 〉〈 math 〉 and 〈 math 〉〈 math 〉, is 〈 math 〉〈 math 〉; there∣fore the second decrement is 〈 math 〉〈 math 〉, &c.) Wherefore the Sum of the Infinite Progression will be to the Sum of the like number of Terms equal to the greatest, as 〈 math 〉〈 math 〉, &c. to 3, that is, by Consect. 3 and 8, as 1 to 3. Q.E.D.

XI. The Sum of a triplicate Arithmetical Progression (i. e. ascending by the Cubes of the Cardinal Numbers) proceeding from 1 thro' 27, 64, &c. ad Infinitum, is Subquadruple of ••he Sum of the like number of Terms equal to the greatest. For the Sum of 4 Terms, 1, 8, 27, 64, i. 100, to 4 times 64, i. e. 256 (dividing both sides by 64) will be found to be as 1+½+〈 math 〉〈 math 〉 to 4; but the Sum of 8 Terms, 1, 8, 27, 64, 125, 216, 343, 582, i. e. 1296 to 8 times 512, that is, 4096 (dividing both Sides by 1024.) will be found to be as 1+¼+〈 math 〉〈 math 〉 to 4, &c. The adhering Fractions thus

constantly decreasing, the one by their ½ part, the others by •••• (for 〈 math 〉〈 math 〉 is 〈 math 〉〈 math 〉, and 〈 math 〉〈 math 〉 is 〈 math 〉〈 math 〉, &c. Wherefore the Sum of the In∣finite Progression will be to the Sum of a like (Infinite) num∣ber of Terms, equal to the greatest, as 〈 math 〉〈 math 〉, &c. 〈 math 〉〈 math 〉, &c. to 4; that is, by Consect. 3 and 8, as 1 to 4. Q. E. D.

XII. The Sum of an Infinite Progression, whose greatest Term is a Square Number, the others decreasing according to the odd numbers 1, 3, 5, 7, &c. is in Subsesquialteran Pro∣portion of the Sum of the like number of equal Terms, i. e. as 2 to 3. For the Sum of three such Terms, e. g. 9, 8, 5, i. e. 22 to thrice 9, i. e. 27. is (dividing both sides by 9) 〈◊〉〈◊〉 2 〈 math 〉〈 math 〉, viz. 〈 math 〉〈 math 〉 to 3, or 2+½−〈 math 〉〈 math 〉 to 3. But the Sum of s•••• such Terms, 36, 35, 32, 27, 20, 11, i. e. 161, to six time 36, i. e. 216 (dividing both sides by 72) is as 2+¼−〈 math 〉〈 math 〉, &c. the adhering Fractions thus always decreasing, some by ½, o∣thers by ¾, as above in Consect. 10. Wherefore the Sum of the Infinite Progression will be to the Sum of the like number of Terms equal to the greatest, as 〈 math 〉〈 math 〉, &c. to 3, i. e. by Con¦sect. 3 and 8, as 2 to 3. Q E. D.

SCHOLIUM II.

THus we have, after our method, demonstrated the chie•• Foundations of the Science or Method, or Arithmetick •••• Infinites, first found out by Dr. John Wallis, Savilian Professo•• of Geometry at Oxford, and afterwards carried further by Det••∣lerus Cluverus, and Ismael Bullialdus. And from these Founda∣tions we will in the following Treatise demonstrate, and that directly and à priori, in a few Lines, the chief Propositions o••

Geometry, which the Antients have spent so much labour, and composed such large Volumes to demonstrate, and that but in∣directly neither.

Proposition XXII.

THe Powers of Proportionals whether continuedly or discretely, such as the Squares, Cubes, &c. are also Proportional.

Demonstration.

Continual Proportionals.	Discrete Proportionals.
a ea e{powerof2}a e{powerof3}a	a ea b eb
Squares aa e{powerof2}a{powerof2} e{powerof4}a{powerof2} e{powerof6}a{powerof2}	a{powerof2} e{powerof2}a{powerof2} b{powerof2} e{powerof2}b{powerof2}
Cubes a{powerof3} e{powerof3}a{powerof3} e{powerof6}a{powerof3} e••a{powerof3}	a{powerof3} e{powerof3}a{powerof3} b{powerof3} e{powerof3}b{powerof3}

Q E. D.

SCHOLIUM.

YOu founded in this Truth, 1. the Reason of the Multipli∣cation and Division of Surd Quantities: For since from the Nature and Definition of Multiplication, it is certain, that 1 is to the Multiplier as the Multiplicand to the Product (for the multiplicand being added as many times to it self as there are Units in the Multiplier, makes the Product) if the √5 is to be multiplied by √3, then as 1 to the √3, so the √5 to the Product; and, by the present Proposition, as 1 to 3, so 5 to the □ Product, i. e. to 15. Wherefore the Product is √15; and so the Rule for Multiplying Surd Quantities is this: Multiply the Quantity under the Radical Signs, and prefix a Radical Sign to the Product.(α) 1.5 Likewise since it is certain from the Nature of Division, that the Divisor is to the Dividend as 1 to the Quotient (for the Quotient expresses by its Units how many times the Divisor is contained in the Dividend) if the √15 is to be divided by √5, you'l have √5 to the √15 as 1 to the Quo∣tient, and, by the present Scholium, 5 to 15, as 1 to the □ of the Quotient, i. e, to 3. Therefore the Quotient is the Root of 3, and so the Rule of dividing Surd Quantities this; viz.

Divide the Quantities themselves under the Radical Signs, and pre∣fix the Radical Sign to the Quotient.

II. Hence also flows the usual Reduction in the Arithme∣tick of Surds, of Surd Quantities to others partly Rational, and on the contrary, of those to the form of Surds, e. g. If you would reduce this mixt Quantity 2a√b, i. e. 2a multiplied by the √b, to the form of a Surd Quantity; which shall all be con∣tained under a Radical Sign; The Square of a Rational Quan∣tity without a Sign 4aa, if it be put under a Radical Sign, in this form √4aa, it equivalent to the Rational Quantity 2a; but the √4aa being multiplied by √b makes √4aab. by N•• 1. of this Scholium. Therefore √4aab is also equivalent to the Quantity first proposed 2a√b. Reciprocally therefore, if th•• form of a meer Surd Quantity √4aab, is to be reduced to on•• more Simple, which may contain without the Radical Sig•• whatever is therein Rational, by dividing the Quantity com∣prehended under the sign √ by some Square or Cube, &c. as here by 4aa, (i. e. √4aab by √4aa, i. e. 2a) the Quotient wil•• be √b, which multiplied by the Divisor 2a, will rightly ex∣press the proposed Quantity under this more simple Form 2a√•• Which may also serve further to illustrate the Scholia of Prop. 7. and 10.

Proposition XXIII.

IF there are four Quantities Proportional, (a, ea, b, eb) they will be also Proportional,

• 1. Inversly. ea to a as eb to b.
• 2. Alternatively,(α) 1.6 a to b as ea to eb.
• 3. Compoundedly,(β) 1.7 a+ea to ea, so b+eb to eb.
• 4. Conversly, a+ea to a as b+eb to b.
• 5. Dividedly,(γ) 1.8 a−ea to
  • ea as b−eb to
  • or a
    • ...eb
    • or 〈◊〉〈◊〉
• 6.(α) 1.9 By a Syllepsis, a to ea as a+b to ea+eb.
• 7. By a Dialepsis, a to ea as a−b to ea−eb.

Which are all manifest, by comparing the Rectangles of the Means and Extremes according to to Prop. 19. and its Consect. 1.

or by dividing any of the Consequents by their Antecedents, ac∣cording to Def. 31.

Proposition XXIV.

IF in a(β) 1.10 double Rank of Quantities you have

• as a to ea,
• so b to eb,

and also

• as ea to oa,
• so eb to ob,

then you'l have also by proportion of Equality orderly pla∣ced,

• as the first a, to the last oa, in the first Series;
• so the first b, to the last ob, in the second Series.

Which is manifest from the Terms themselves.

Proposition XXV.

BUt(γ) 1.11 if they are disorderly plac'd

• as oa to ea
• † so eob to ob

* as ea to a † so ob to eb, * you'l have here again by proportion of Equality,

• as the first oa to the last a, in the first Series;
• so the first eob to the last eb, in the second Series.

As is evident from the Rectangles of the Extremes and Means, as also from the very Terms.

Proposition XXVI.

IF(α) 1.12 as the whole ea to the whole a, so the part eb to the part b; then also will

the Remainder	Remainder	Whole	Whole
ea−eb to the	a−b, as the	ea	to the a.

This is evident from the Rectangle of the Extremes and Means, both which are eaa−eab. Q.E.D.

Proposition XXVII.

REctangles or Products having one common Efficient or Side, are one to another as the other Efficients or Sides.

Demonstration.

Suppose the Products to be ab and ac, having the common Efficient a; I say they are

• as b to c, so ab to ac.

Which is evident at first sight, by comparing the Products of the Extremes and Means, and also fully shews, that other way of proving Proportionality, whereby by dividing the Conse∣quents by their Antecedents, the identity or sameness of the Quotients are wont to be demonstrated.

SCHOLIUM I.

I. THe Reduction of Fractions either to more compounded or more simple ones is founded on this Theorem; on the one hand by multiplying, on the other by dividing, by the same quantity, both the Numerator and the Denominator, as, e. g, 〈 math 〉〈 math 〉 and 〈 math 〉〈 math 〉 and 〈 math 〉〈 math 〉, ⅓, 〈 math 〉〈 math 〉, 〈 math 〉〈 math 〉, &c. are in reality the same Fractions. And

II. The Reduction of Fractions to the same Denomination, as if 〈 math 〉〈 math 〉 and 〈 math 〉〈 math 〉 are to be changed into two o∣thers that shall have same Denominator;(α) 1.13 this is to be done by multiplying the Denominators together for a new Denominator,(β) 1.14 and each Numerator by the Denominator of the other for a new Numerator, and you'l have for the two Fractions above — 〈 math 〉〈 math 〉 and 〈 math 〉〈 math 〉

SCHOLIUM II.

WE will here for a conclusion of Proportionals, shew the way of cutting or dividing any Quantity in Mean and Extreme Reason, viz. if for the greater Part you put x, the less

will be a−x; and so by Hypoth. these three, a, x and a−x, will be proportional, by Def. 34. Therefore by Prop. 17. the Product of the Extremes aa−ax = to the Square of the Mean xx, and (adding on both sides ax) aa=xx+ax; and more∣over adding on both sides ¼aa, you'l have 〈 math 〉〈 math 〉aa=xx+ax+¼aa. Now this last Quantity, since it is an exact Square, whose Root is x+½a, you'l have √〈 math 〉〈 math 〉aa=x+½a, and (subtracting from both sides 〈 math 〉〈 math 〉 a) √〈 math 〉〈 math 〉aa−½a=x.

Now therefore we have a Rule to determine the greater part of a given Quantity to be divided in Mean and Extreme Rea∣son, viz. if the given Quantity be a Line, e. g. AB=a (Fig. 58.) join to it(α) 1.15 at Right Angles AC=½a: Wherefore by the Theorem of Pythagoras from Schol. Definit. 13. the Hypo∣thenuse CB, or, which is equal to it, CD=√¾aa; and conse∣quently AC=½a being taken out of CD, the Remainder AD, or AE, which is equal to it, will be = x, the greatest part sought; according to Euclid, whose Invention this first Specimen of Analysis, by way of Anticipa∣tion, reduces to its original Fountain. As for Numbers (tho none accurately admits of this Section) the sense of the Rule, or which is all one as to the thing it self, is this: Add the Squares of a whole Num∣ber and its half, and subtract the said half from the Root of the Sum (which can't be had exactly, since it is √〈 math 〉〈 math 〉.

Notes

• (α) 1.1

  Eucl. 17 l. 6 & 20 l. 7.
• (β) 1.2

  lib. I. de Sphaer. & Cyl. Prop. 14.
• (a) 1.3

  lib. I. de Sphaer. & Cyl. prop. 15.
• (b) 1.4

  E••cl. 16 l. 6 & 19. l. 7.
• (α) 1.5

  Eucl. lib. 6. prop. 22.
• (α) 1.6

  Eucl. 15, 16. v. 9 10, 13, vi••.
• (β) 1.7

  18, v.
• (γ) 1.8

  17, v.
• (α) 1.9

  Eucl. 15, 16. v. 9 10, 13, vi••.
• (β) 1.10

  Eucl. 3, 20, 22. lib. v. 14. vii.
• (γ) 1.11

  Eucl. 21, 23. lib. v.
• (α) 1.12

  Eucl. 1. 12. v. 5. 6. 12. vii.
• (α) 1.13

  Eucl. 5 & 19. lib. v. 7 & 11. l. 7.
• (β) 1.14

  Besides se∣veral other Prop. see also the 17 & 18 lib. vii.
• (α) 1.15

  Eucl. 11. lib. II. & 30. lib. vi.