as above) =

21-2+dn Multiply both by md, and it is nnd=2ln—2s+dn;

nd

add 25, and then fubtract nnd from both Sides, and we have 2s=2/ nd n―n nd ̧ 21+d=6, then it is 25=bn—w2;

Divide all by d, and it is

xn-m2. Call 21+d

b

b2

the Rule for # when / is

and (by Prob. 6. Chap. 2. §. 2. Book III.) n= + given.

SCHOLIUM. These Rules are tedious both in the Investigation and Application: But there is another Method of folving the Problem, which tho' it is only by Trials, yet it proceeds directly and certainly to the Anfwer; and is rather eafier than the former Work, and therefore I fhall here explain it.

Another RULE.

By Prob. 1. raise a Series from the given Extreme and Difference; and take the Sum of the Series gradually as it rifes, continuing this Operation till the Sum is equal to the given Sum; and the Series fo raised will fhew both the Number of Terms, and the Extreme fought; which is the last Term found in the Series.

4+3+3+3+3+3+3 4. 7. 10. 13. 16. 19. 22 4. II. 21. 34. 50. 69. 91

The Reason of this Rule will be obvious from one Example. Suppofe a=4, d=3, S=91. In the annext Operation, you fee in the upper Line the given Extreme 4, and the Difference 3 continually added. In the fecond Line are the Terms of the Progreffion formed by that continual Addition; and in the third Line are the feveral Sums of the preceding Series taken continually from the Beginning, by adding the next Term to the preceding Sum. Whence we fee in the prefent Example, that the Extreme fought is 22, and the Number of Terms 7.

Obferve, That the Tedioufnefs of this Method, when the Number of the Terms is great, may be relieved by the following Means, viz. Take any Number for at a guess, (in which to prevent being too wide of the Truth, have a regard to the given Numbers); then by this Number n, with the given d and a or 1, find the other Extreme, and the Sum; and if this Sum differs from that given, begin at the Extreme laft found, and raise a Series, increasing or decreafing, as the Cafe requires, till you find a Sum equal to the given one. For Example: Suppofe a = 3, d = 4, s=406; I guefs = 12, and hereby find /=47 (=4X 11+3) and s= 300 (=47+3×6); which being lefs than 406, I

47+4+4 51. 55 300.351.406

begin at 47, and adding the Difference 4 till the Sum is equal to 406, I find that this happens upon adding that Difference twice, i. e. that two Terms more with the 47 make the Sum given; whence 'tis certain that 14 is the Number of Terms, (for there were 12 to bring it to 47, and 2 now added) and 55 the greater Extreme.

SCHOLIUMS.

I. For the more convenient and ready Use of the last seven Problems, we fhall put them all in a Table, that they may appear in one View; expreffed fimply by their Characters, whofe Signification I fhall repeat:

leffer Extreme. 1=greater Extreme.

"Number of Terms.

d the common Difference.

s Sum of the whole

Series.

The Solution of this is by raifing a Series from the given Extreme with the given Difference, till the Sum is equal to that given.

25+ d n2 = dn

27

a

If. According as the given things in any of the preceding Problems are chofen or related to another, fo will the Problem be offible or impoffible: For any three Numbers taken at random will not make a poffible Problem of each of them, (tho' of fome it will); because there are particular Relations, within certain Limits, which the given Numbers must have to one another in most of thefe Problems; fo that they may be poffible, i. e. fo that the three given things may belong to the fame Progreffion. The Poffibility or Impoffibility will appear by applying the Rules; for if the given Numbers are inconfiftent, one or more of the things fought will be found impoffible, by fonte Abfurdity that will appear in applying the given Numbers to one another according to the Rule.

But now if you require how to invent three Numbers confiftent with one another, to make Data for any of thefe Problems; it may be done either of these Ways; viz.

r. Take any two Numbers whatever for a, d, and any Integer greater than 1, for "; and by these three find s and thus you have five things all belonging to one Progreffion, out of which to chufe any 3 for Data of a Problem. The Reafon of this Rule is plainly thus: That from any Number a we may raise a Progreffion with any Difference d to any Number of Terms n we pleafe.

2. Or take any two Numbers for a,1, fo that a do not exceed 7; and any Integer greater than form, and by these find d, s. The Reafon of this is, that betwixt any two Num

bers a,1, any Number of Arithmetical Means may be placed, as has been fhewn in Corol. to Prob 5.

3. Therefore if a,n,d, or a, n, I are the Terms to be invented, we can find them by themselves; and if any 2 of these 3, (as a, n. a, d n, d. a,l. n, l. d,l.) with any other Term [except that one Cafe d,s] are to be invented, we can find them without finding all the 5; yet one of the two things not required must be found: for we must take either a, n, d, or a, n, l, and by them find the other Term to be invented. But if d, and s are to be invented, we must find all the 5 by means of a, n, d, or a,n,l.

But again, it may be required to invent the three things to be given in each of these Problems, without the Invention of any of the other two; which by the Rules now given cannot be done, except when a, n,d, or a,n,l are to be given. For this you have Rules in what immediately follows, when it is poffible to be done.

III In the preceding Problems, no less than three things are neceffary to be known, to make each of them determinate to one certain Solution: But if we fuppofe only two of the five things to be given for finding the other three; then, of fuch Problems, fome will be indeterminate, and have an infinite Number of Solutions, i. e. we can find an endless Variety of Numbers for the three things fought, which will all fatisfy the Problem: Alfo of thefe indeterminate Problems, fome will be abfolutely indeterminate as to fome of the things fought, fo that any Number whatever may be affumed. But others of the things fought [and in fome of thefe Problems all the things fought] must be taken within certain Limits, which nevertheless admit of an infinite Variety of Solutions. Again; Others of thefe Problems, wherein only two things are given, will be determinate to a certain Number of Solutions, according to the different Circumstances and Relations of the two given things; for there will be one or more Solutions, as thefe Circumstances differ.

Of all thefe Problems, Indeterminate or Determinate, there are juft 10; because there are just so many different Choices of two things to be found in five. Thus; the five things being a, l,d, n, s, the Choices of 2, to make Data of a Problem, are thefe; a,l. a, d. a, 2. a,s. l, d. l,n. 1, s. d,n. d,s. n,s. Of which there are 6 that are Indeterminate, and 4 Determinate.

J. 3. Containing Problems concerning Arithmetical Progressions, wherein two things only are given to find the other three.

THA

HATI may deliver the Rules and Demonstrations of the following Problems in the moft fimple and easy manner; and that you may understand them aright, take these few previous Explanations.

1. Tho' I have fhewn in moft of the following Problems, how to find, by means only of the two given things, any one of the three things fought, you are not to understand it, as if all these three Rules were to be applied in the fame Solution, i. e. as if three Numbers found, one by each of thefe Rules, might be taken for the Solution of the Problem; because there being a Variety of Solutions for each of these three things, any one Solution for each of them will not make a Combination that can folve the Froblem; for this plain Reafon, viz. when any one right Number is taken for any one of the three unknown things, this with the two given things determine the other two things fought, according to the preceding Problems, which have but one limited Solution; fo that we cannot with any one Solution, for one of the things fought, join any one of the Solutions for the other two things, thefe being now deterinin'd by the Solution which we have chofen for the former one, together with the two given things: therefore the particular Rules for the different things fought, are to be understood only as Steps in fo many different Methods of folving the fame Problem; which are to be applied thus: By the two given things find any one of

Ii 2

the

the unknown things, according to the Rule given for it; then take the thing now found, with the two given things; and by thefe three find the other two things fought, by that one of the preceding Problems where these three things are given: But the Rules for these are fet down along with the other.

2. As any three things taken at random could not make the preceding Problems poffible, fo neither here will any two Numbers make any of the following Problems poffible; there being certain Limitations, in respect of one another, under which they must be taken in fome Problems, tho' not in all. That we may not encumber the Problems with these things, I fhall here explain these Limitations where they ought to be, and shew where there are none. Thus:

a has no Limitation, and may be any Number whatever, and even o.

I may be any Number whatever, if it's not less than a or d, nor greater than s; but has no Limitation with respect to n.

d may be any Number whatever not exceeding or s, or it may also be o; but has no Limitation with respect to a or n.

must be an Integer greater than 1; and tho', ftrictly speaking, there is no Progreffion without three Terms, yet we fhall here allow of two Terms as the smallest Progreffion. s may be any Number whatever, not less than a, or d, or l; but has no Limitation with refpect to n.

The Reafon of these things is obvious from the Nature of a Progreffion, which may begin with o, or any Number, and proceed by any or no Difference;. and may confift of two only, or any other Number of Terms.

Now in all the following Problems it's fuppos'd that the two given things are confiftent, according to these Directions; and for the Limitations expreffed in the Rules for finding one of the things fought, they do not only comprehend the Conditions now mention'd, but fome of them contain more ftrict ones, because regard is to be had to the rest of the five things. And in the Reafon given for thefe Rules, the general Conditions now mention'd are frequently fuppofed; which therefore must be always in view, because fuch fimple things need not be repeated, unless where there is any danger of Obscurity.

3. As to the Method of investigating the following particular Rules, I shall here give you a general Account of it, that I may fave the repeating of the fame things, as otherways would be neceffary in the demonftrating of each of them.

25

12 — a2

In the first place, they depend upon the preceding Problems, and are discovered thus: I take that one of thefe Problems in which are given the two given things of the prefent Problem, and that one of the three things fought that I would now firft find; then by the Rules of that Problem for finding the remaining two things, I difcover what Limitations that one I would now find lies under with refpect to the two given things, that these two remaining things may be poffible; and then I conclude, that the thing I feek being taken within this Limitation, it belongs to the fame Progreffion with the two given things; and is therefore a true Solution. But more particularly: Suppofe a, l are given, and I would find s; I go to the preceding Prob. 6. wherein a, l,s are given; and there the Rules for finding n,d are these, n= And here to make n poffible, it's plain that 25 muft be Multiple of a+1, because n must be an Integer greater than 1; and the fame Condition of s will make d poffible; for fince a does not exceed 1, there is nothing to make it impoffible, but at being greater than 2s; which it is not, if 25 be a Multiple of a-+. Therefore if s is taken fo, as 2s be a Multiple of a+ [or alfo if we take for s any Multiple of a+1; from whence certainly follows that 2s is a Multiple of a + any fuch Number folves the Queftion for s; that is, s fo taken with a,, belong to the fame Progreffion. For if they did not, n, d could not be found by their Means, according to fuch Rules as have been difcovered and demonftrated upon that very Suppolition, that all the five things do belong to the fame Progreffion.

and d=

25-a

This

This manner of drawing the Conclufion you are to fuppofe in all the following Demonftrations, which I fhall never again repeat; but only thew you how that the Number fought being taken according to the Rule, is confiftent with the Poffibility of the remaining two things to be found. In doing of which, I have frequent Ufe for this Principle, viz. If one Number is equal to, greater or leffer than another, any Multiple, or aliquot Part of the former is alfo equal to, greater or leffer than the like Multiple or aliquot Part of the other; which being fo very fimple, it will be obvious in the Places where I ufe it, and therefore I fhall not again repeat it.

We come now to the Problems, whereof the first fix are Indeterminate, and the other four Determinate; and mind, that the Problems here referred to, are thofe of the preceding §.

PROB. XII. Given a, 1, to find n, d, s.

1. For n. Take any Integral Number greater than 1; you have the Reason of this in the preceding Scholium 1.

2. For d. Take any Number fuch, that be an Integer, i. e. take any aliquot Part of l—a. For by Prob. 6. n=2=a+1, and s= al+di+k—a2. Which are both

2 d

poffibile, as d is taken; because » being an Integer, must be so; and if we divide l—a

by any Integer, and call the Quote d, the fame d dividing the former Integer; therefore any Number d, which is an

1-a

a will return for a Quote aliquot Part of I-a, makes

an Integer. As for s, it requires only that be not less than a, which is the general

Condition.

3. For s, take any Multiple of a-+-1, or the half of any fuch Multiple. For by Prob. 8.

12 — a2 25-a

which are both evidently poffible, as s is limited.

PROB. XIII. Given n, d, to find a, 1, s.

1. For a, take any Number, or even o. The Reason is fhewn already.

and s=

2

2. For, take any Number not lefs than dn1. For by Prob. 7. a=l-dn—1, 2ln-dn2+dn. The former contains the very Conditions of the Rule, and the other requires only that 21n+dn be greater than dn2, or al+d greater than dn. But if / is at leaft=dn-d, then is 21+d=2dn−2d+d=2dn-d; which is greater than dn, because z is at least 2.

3. For s, take any Number greater than

dnz. dn

2

and /=

25+ dn2 — d n

Now if s is

than greater

For by Prob. 10. a 25-dn2+dn dn2-dn, then is as greater than

2

dn2—du, and 2s+dn greater than d2; which evidently makes a and 7 poffible.

PROB. XIV. Given a, n, to find 1, d, s.

1. For 1, take any Number not less than a; the Reason is fhewn above. 2. For d, take any Number greater; the Reafon is also fhewn above.