PRO

431

P li O

2, 3, 4, Sec. are -j-, 4-, ^. &c. which arc hannonicals ; for

— : - 1 -- ; and so on. And, in general, the

reciprocals of thearilhmelicals a, a-\-d, a-\-2d, a-{-3d, &c. viz.

-—;-„ —■—&c. are hannonicals ; et b contra. 2. If

<H-d a + 2d a + 3d

three or four numbers in .harmonica! proportion be either multi-plied or divided by some number, the products, or the quotients,will still be in harmonica! proportion. Thus, the hannonicals 6,8, 12, multiplied by 2, give 12, lb, 24, or divided by 2, give3, 4,6, which are also harmonicals. 3. To find a hannomcnl meanproportional between two terms : divide double their product bytheir sum. 4. To tind a third term in harmonica! proportion totwo given terms; divide their product by the difference betweendouble the first term and the second term. 5. To tind a fourthterm in harmonical proportion to three terms given: divide theproduct of the first and third by the difference between double thefirst and the second term. Hence of the two terms a and b the2 ab

harmonical mean is —j-^ ; the third harmonical proportion is

ab

also to a , b , c , the fourth harmonical is

271-/).. .. 2 u - b ' U

there be taken an arithmetical mean and a harmonical mean, be-tween any two terms, the four terms will be in geometrical pro-portion. Thus, between 2 and 6 the arithmetical mean is 4, andthe harmonical mean is 3 ; and hence 2 : 3 : : 4 : 6. Also be-et-)-/)

tween a and b the arithmetical mean is ■

and the harmoni-

cal mean is

2 ab

b.

, , ?«/) u

’ —i—7 , but u : —— ; : ; -n-\-b a-\-b

Proportion, Inordinate, is where the order of the termscompared is disturbed or irregular t e. gr. in two ranks ol' num-bers, three in each rank, viz. in one rank 2, 3, 9 ; and in the other,8, 24, 36; which are proportional, the former to the latter, but ina different order, viz. 2: 3 : : 24 : 36, and 3 : 9 : : 8 : 24. Thencasting out the mean terms in each rank, it is concluded that 2 :9 :: 8 : 36, that is, the first is to the third in the first rank, as thefirst is to the third in the second rank.

Proportion of Figures.—Prob. I. To find the proportionthat one rectangle hath to another, both length and breadth mustbe considered. For rectangles are to each other, as the productsof their respective lengths multiplied by their breadths. 1. Titus,it there be two rectangles, the former of which hath its length fivefeet, and its breadth three ; and the latter hath its length eightfeet, and its breadth four. Then the rectangles will be to eachother as 3 X 5 t = 15), is to 4 x 8 (—32) that is, as 13 : 32, sothat all rectangles are to one another in a ratio compounded ofthat of their sides. 2. When rectangles have their sides propor-tionate, so that AB : EH :; AD : EF, (Plate CXXXVH. fig.12,) then is the rectangle A, to the rectangle B, in a duplicateproportion to the ratio of the sides. For the ratio of A to B, iscompounded ©f the ratio of AB to EH, and of the ratio of AD to1\F. And therefore the proportion of A to B, being compounded°t equal ratios, must he duplicate of the ratio of their sides to eachother ; that is, duplicate of the ratio of AB : EFI, or of AD : EF.Hence all triangles, parallelograms, prisms, parallelopipeds, py-famids, cones, and cylinders, are to one another respectivelycompared, in a proportion compounded of that of their heightsant. bases. 3. All triangles, and parallelograms, pyramids, prisms,'and parallelopipeds; also all cones, and cylinders, each kitul com-pared among themselves ; if they have equal altitudes, are in thesame proportion as their bases; if they have equal bases, are asme *°. r the bases, or heights, will severally be common

f ,, cu ' nts or multipliers; and therefore must make the products ben tiie same proportion as the multiplicand was before. Thus, if the

equal altitude of any two triangles, parallelopipeds, cones, &c. becalled A, and their unequal bases B, aud D : then it wil be as B :D : : AB : AD. This problem being of great use in geo-metry.

Case 1. To find two right-lines, whose sum or difference is given,reciprocally proportional to two given lutes. Let the two givenlines be B and C, and let the sum of the two lines sought be Z :it is required to find a point where Z may be so divided, as that 15: A : : E ; C, (tig. 13.) First find m, a mean proportional betweenB and C, which erect perpendicularly at either end of Z ; drawD parallel to Z, describe a semicircle upon Z, and where D cutsthat semicircle, let fall m perpendicularly : then A and E are thelines required : for A x E — «('- = BxC. 2. E. D.

Case 2. When the diifetjence — X is given, (fig. 14,) find m amean proportional as before, which erect perpendicularly at eitherend of X : thus draw R from the middle-point of X to M, andwith that as radius descri'oe a semicircle on the centre C : so shallA -f- X, be the greater line, and E the less sought, for (A-)-X)X E = m 2 = BC. 2. li. D.

Pros. II. Having two squares, to find two others reciprocallyproportional, whose sum is equal to a given square. Let thesquares given be/)-, c 2 , and the reciprocal required y 2 and d- — y 2

then,

y -: b 2 :: c -: d ? — tj 2d-if l — )/' = bh - 1y i _ d'y + jd * = 1 dl -id 2 — y z = y U& — lr'-c 2

y % — -‘rf V — b '‘ C 2

y =\ / TI i

Vc 2

' Vb-e-

The construction of this is as follows, let AB, (fig. 15,) and AC —b,find a fourth proportional, as C E = and on CF

and, BD = c

=-td describe a semicircle, and in it apply CG = CE, then F G

will be equal

Vi d l —bH-*

d

Let IIC = d, and Cl

d-

V\d* — b z c z , the mean proportional will be CK = y, and onCll = d describe a semicircle, and in it apply CL=CK, thenLI1 = Vd l — y z , which is the side oi the other square sought.

Prob. HI. Having two squares, to find two other reciprocalswhose difference shall he equal to a given square. Let the squares

given

then,

be J -, g 2 , and the reciprocals sought ij l and h- -f- y 2 t

yi -|_ hy — j !„2

y ' -+• + y ? = f * g 2 + ih >

y z -f jh-_= \ /j y y I/,I

lk 1

T ==.V/V+ jh*-_ _

y=\ Vjy ~jl i /r»__ i/ t ».

The construction of this is almost the same as in the precedingproblem. Or the proportion of solids, every parallelepiped, is toa pyramid of the same base and height, as 3 to 1, that is, the one istriple the other. A cylinder, spheroid, and cone, of the same baseand height, are as 3, 2, and 1.

PROPORTIONALS, in geometry, are quantities either linearor numeral, which bear the same ratio or relation to each other.

PROPOSITION, m logic, pail of an argument wherein somequality, either negative or positive, is attributed to a subject. SeeLogic .

Proposition, in mathematics, is either some truth advancedand shewn to be such by demonstration, or some operation pro-posed, and its solution shewn. If the proposition be deduced fromseveral theoretical definitions compared together, it is called atheorem; if trom a praxis, or series of operations, it is called aproblem. See Geometry .

Proposition, in oratory 7 . See Oratory.

Proposition, in poetry, the first part of a poem, wherein theauthor proposes briefly, and in general, what lie is to say in thebody of his work, It should comprehend only the matter of thepoem, that is, the action and persons that acts.

7 PROPJREFEGT,

I