PRO
of the prtgrefim, when it is fuppofed to be continued infinite- ly. Thefe limits are analogous to the limits of figures, and they mutually aflift each other. The areas of figures can, in many cafes, be no othenvife exprefled than by fucn pro- grtffims; and when the limits of figures are known, they may fometimes be advantageoufly applied for approximating to the fums of certain vrogreflims.
Thus, for inftance, let the terms of any P™grf'« 'be repre- fented by the perpendiculars A F, B E C K, H L, &c. Band- ing at a given diftance on the bafe AD; and let P N be any ordinate of the curve, FN, palling over the extremities of thofe perpendiculars. Suppofe A P to be produced : then ac- cording as the area, A P N F, has a limit to winch ,t may ap- proach continually, but never exceed, or may be produced till it exceed any given fpace; there will alfo be a limit of the fum of the progrcffion, or it may be continued till it exceed any given number. For fuppofmg the reflaiigles F B, E C,
i \v
PRO
KB, LI, &c. completed, the area A P N F will always lefs than the fum of thofe rectangles, but greater than their fum after the firft. Therefore the area A P N F, and the fum of thofe rectangles, either both have limits, or both have none. The fame is to be faid of the fum of the ordinates A F, BE,CK,HL, &c. and of the fum of the terms of the m peprefented by them. If the curve, F N *, for ex- be the common hyperbola, b its center, b P the afymp- ■ I A B being equal to b A, if A F reprefent unity, the feries of ordinates will reprefent the progrejjion i, £, $1 i, i- h ,&c. which may therefore be continued till it exceed any given number, as the hyperbolic area may be produced till it ex- coed my given fpace*. Put if FN* be an hyperbola of any highei order, fo that the ordinate P N be reciprocally as any power of the bafe b P, whofe exponent is greater than unity, the area A P N F b , and the fum of the program re- prefented by the feries of ordin tes, will have limits Hence there is always a limit of the fum of the fractions that have unity for their common numerator; and the fquares, cubes, or any other powers of the numbers, i, 2, 3, 4, &c. whofe exponents exceed unity for their fucceflive denominators c . [» See Hyperbola. b Ibid. c See Madaurbfc Fluxions,
Hi. p. 290.]
When the area APNF has a limit, we may not only con- clude, that th:- fum of the progrejjion reprefented by the ordi- nates has a limit; but when the former limit is known, we may by it approximate to the value of the latter: and vice vsrfa, when the limit of the pngrejftm is given, the limit of the area may be found. See Madaurhy Lib. cit. Art, 152, 353.
Prcgreffims of fractions may be found at pleafure, that {hall have -imgnable numbers equal to the limit of the fum of the | terms. Thus a feries or progreffm of any number of quanti- : ties continually dteereafihg being given, their fucceflive differ- I ence's form a new feries of terms, the fum of which from ■ the beginning is always equal to the excefs of the firft term of 1 the firft feries" above i'ts laft term. For inftance, if A, B, C, j D, E, £cc. be the terms of the firft feries, it is manifeft that] the fum of the difference of A and B, B and C, C and D, D and E, is the excefs of A above E. If the terms of the I firft feries decreafe in fuch a manner, that by continuing the progreffien they may become lefs than any quantity that can be affigned, then the firft term of the firft feries is the limit of the fum of the fecond feries. In like manner, the differ- ence of the alternate terms of the firft: feries, as of A and C, B and D, C and E, &c. form a new progrejfton of terms, the fum of any number of which is equal to the excefs of the fum of A and B, the firft and fecond term of the feries, above the fum of the laft and penultimate terms; and the fum of A and B is the limit of the fum of the new feries. In general, if a progrejjion is formed by taking the difference of the firft term A, and the term whofe place~in the feries is expreffed by any number n; of the fecond term B, and that whofe place is b-I-i; of the third term C, and that whofe place is a + 2, and fo on; then will the limit of the fum of this new pro- grcjfmt be equal to the fum of the terms A, B, C, D, &c. which precede that term whofe place is exprefled by «. In this manner progreffiom may be found at pleafure, which may be continued without end, and have given numbers for the limits of their fums.
For example, let the firft feries be r, i, jt £, -f' & c - the &c- ceflive differences of the terms of which are \. £. rV- re> &c. and the limit of the fum of this pregrejfim will- therefore be 1 .
If we multiply each term of this laft feries by 7, that the firft term may be unit, we fhall have 1, 7. £, -iV- &c. the deno- minators of which are the triangular numbers, unity being the common numerator, and the limit of the fum of this fro- grefpen is 2. The fucceifive differences of the terms of this latter feries being each multiplied by |, that the term of the new feries may be unity, give r, +, ts, -,'^ &c. which have the pyramidal numbers for their fucceflive denominators; and the limit of the fum of this progiejfion is |. In the fame man- ner, the limit of the fum of the fra6tions having unity for their common numerator, and the figurate numbers of any order denoted by m for their fucceflive denominators, is found
to be .
m — 2
The fame feries 1, ^, 4- h r> & c heing again aflumed, the differences of the alternate terms are f, f , T \, -^ &c. the li- mit of the fum of which progrejjion is ir- Dividing each term by 2, the limit of the fum of y. j, -,V or- &c. is \. If we take the differences of the firft term, and that whofe place is to, the fecond term and that whofe place is m-\- 1, &c. the common numerator of thofe differences will be m — 1; and their fucceflive denominators, the produces of iXw, i"Xm\-i* 3XW-J-2, &c. and the limit of the fum of this progreffien is the fum of as many terms i-f-i-f-j-r-,!,, & c - as there are units in th — 1. Now if each term of the new progreffien be di- vided by m — r, that unity may be the common numerator,
the terms — ,
, &c.
rill arife, the!i-
2X«+l' 3X»i4-2'
mit of which is equal to the fum of the fractions i, I, T , \, &c. (continued till their number be fn — 1) divided by m — I. In like manner, by afluming other alternate, or any equiva- lent terms of the feries 1, 5, f j-, &c. we may form new pro- greffiom, the value of which may be found. Thus, if we take the terms 1, \. £, T ' T , &c pafling over three terms, and divide the fucceflive differences of thefe teims by 96, we fhall have
the feries . &c. which is
5- 2d, 5.9 24, 9.I3.24, 13 17.24.
equivalent to the feries C, given by Monf. de Monmort, in the phHofophical tr an factions, N°. 353, p. 651. viz.
c _ _!_ + _Ii— + U— + L12 .
i. 2. 3. 4. 5*5. 6. 7.. 8. 9 ' 9. 1 0.1 1. 1-2. 13 "13. 1 4. 15. 1 6. 1 7. -f- &c. The fum of which will therefore be ~ z . And if we take the alternate terms of the feries i, A,?V. yV> &c- above men 1 '- ' and divide the fucceflive differences of the terms 5 9 13
2.12* 12.30' 30.56*
5 9 13
1.2.34.' 3.4.5.6' 5.6.7.8*
1 , &c. mentioned in the faid philofophical tranfac-
7.5.9.10
tions, and marked A, the fum or limit of which, by the fore- going rule, will be £. So the limit of Mi. Monmort's feries
by 2, we fliall have the feries which is equivalent to the feries
&c.
B:
7+7
&c. will be = See Mac-
where there is an error of the
3-4-5 4-5- D '7'8 lauritfs fluxions, art. 356, prefs, in p. 296. 1 1 1. and 1 5. for the feries which in line 1 r. is faid to be Mr. Monmort's feries B, is the feries A, p. 651. of the philofophical tranfaclions, N 3 . 353 atorefaid; fo in fjnc 1 5, for A, read B,
This may fuffice to fliew how the fums of pro«rejfr,m fo de- rived may be found. We refer for the farther application of thefe principles to the aforefaid trcatife of fluxions, art. 357, &c. Mr. Stirling's 3 treatife of thefummation of feries, ought alfo to be confulted, as he has improved the method of approxi- mating to the value of progreffiom, which often arife in the folution of problems. See alfo Mr. de Moivre's mfcellanea aftalytka, and particularly the fupplement to that work. [*Me- th dus Mfferentialk : five traSiatus de fummatione et interpola- ■ tione ferierwn i fin; t arum. Lond 1730. 4*-'.]
PROGYMNASMATA, npw^f/A, in antiquity, certain preparatory exercifes performed by all tho'e who offered them- felves to contend in the Olympian games. Potter, Archasol. 1. 2. c. 22. T. 1 . p. 448.
PROKIBITO, in the Italian mufic, is a term applied to fuch parts of a' piece as are not proper, or according to juft rule. Thus intervallo probibito is every interval in melody that does not pafs the ear eafily or naturally, to give it fome pleafure; fuch are the tritone, the fixth major, the feventh, ninth, &c. though under certain circumftances, even thefe have pleafing effects, in that by their harfhnefs they render the fubfequent concords more agreeable.
PROJECTILE (Cycl.)-The theory of projcffi'es, as delivered under this head in the Cyclopaedia, and by aim oft all writers on gunnery fince Galileo, proceeds on the fuppofition, that the flight of fhot or fhells is nearly in the curve of a pa- rabola. Galileo, indeed, has fhewn, that independently of the refiftance of the air, all prjeftiies would in their flight de- fcribe a parabola; and did pro; ofe fome means of examining what inequalities would arife from that refiftance. So that it mi<*ht have been expected, that thofe who came after him would have tried how far the real motions of projectiles de- viated from a parabolic tract, in order to havj decided how
fat
