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all they propose is , to determine the velocity wherewith a generated quantity increases , ancl to that has been generated or described by tho

variable fluxion . On these two bases fluxions Here follow two of Mr . Murdoch's instances : 1 st . — -A heavy body descends perpendicularly in a second , aucl at the end of this time has velocity of 32 * 0 feet in a second , which is accuratel iven distance then the bodfelltake

at any g y , the right line , and the velocity of the falling point may be truly computed ; but the velocity above A , at ever so small a distance , will be less and the velocity at any point below A , tit the distancewill be greater than in A

, . It is therefore plain , that in the point A the certain determined velocity which belongs to no in the whole line . Now this velocity is the right line in the point A , and with it the body would if gravity acted no longer on the body ' s arrival

2 nd . —Take a glass tube open at both ends , whose is of different diameters iu different places , and iu a stream till the water fills the tube and it ; then in different parts of the tube the water will be as the squares of the diameters , and

quence different . Suppose then in any marked place a plane to pass tube perpendicular to the axis , or to the motion of and of consequence the water will pass through with a certain determined velocity . But if anoth be drawn the formerthe water

ever so near , , the different diameters , will flow through this with different from what it did at- the former ; and one section of the tube , or single point onl y , the velocity belongs . It is the fluxion of the which the fluid

space that section , and ivith that uniform velocit y the continue to move , if the diameter was the same of the tube . 3 rd . —If a hollow cylinder bo filled with water freelout through hole at the bottomthe

y a , effluent will bo as the height of the water ; and surface of the incumbent fluid descends without velocity of the stream will decrease , till the out . There can then be no two moments of time each other over so nearlywherein the velocity of

, is the same ; and of consequence the velocity at point belongs onl y fco that particular indivisible time . Now this is accurately the fluxion of the flowing ; and if , at that instant , more water was the cylinder to make the surface keep its place ,

would retain its velocity , and still be the fluxion Such are the operations of nature , and they visibl the nature of fluxion . It is from hence quite clear thafc the fluxion rated quantity cannot retain any one determined

the least space of time whatever , but the moment at that value , the same moment it loses it fluxion of such quantity can only pass graduall cessivel y through the indefinite degrees , contained tlie two extreme values , which are the limits thereof the of the fluentin the fluxion

generation , case -But then , though a determinate degree of fluxion continue at all , yet at every determinate indivisible of time , every fluent has some determinate degree whose abstract value is determinate in itself ; fluxion has no determined value for the least space

whatever . To find its value then , that is , the fluxion has to anoher , is a problem strictly geometrical Avithstanding anti-mathematicians have declared the Mr . Murdoch ' s was a most ingenious and new determinating expeditiousl y the tangents of

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or flowing , sum up all continuall which a mathematical reader often lus in the common way ; and as the tangents of is of tho greatest

y . 1 G-12 feet acquired a y known ; in curves minations exhibit the gradations of easy method in doing the thing , is a in the best manner . The rule is this : —Suppose B D abcissa = x , C D the ordinate == y , and the nature of the be such

point A y in the any point than in A , possible curve of y ordinate be ou one side of the x x y- \ -xyy - « + a ay — a a 4- a , greatest power of y be wanting , the

body has a other point of that proceed , A . concavity

immerse it through of the of consethrough the

the water , this section er section reason of a velocity therefore to Then make a fraction and a numerator by taking all the terms wherein the all their signs , and if the known quantity sion , to prefix unity ; and if two , 2 ; if will have 3 3 j 2 2

determinate describes at fluid would to the end — a -- a a y — a x x , The fraction , by assuming the terms occurs , and retaining the . signs , ancl if one dimension , to prefix unity , as above be 3 ar' — 2 x x y + x yy — a a -j- 2 a of these by x , aud the denominator

, to flow y of the since the stop , the be all y // — a a -f 2 a x . This fraction is equal to AB , and , _ — 3 of + 2 aay — 2 act , x -f 3 xx — 2 x y x y y — a a + 2 In this easy way may the tangents be exhibited ; and I addby the same

succeeding tlie water any given moment of fluid then poured into , be skilful , may the tangents of infinite determined . VOICES FKOM MOST people are probably aware ot

effluent ofthe fluid . y confirm a genevalue , for out Ireland of a number of ancient from , their form ordinarily called " the learned have named them variousl " fire towers , " " ivatch towers , " " tower which names tire referable respectivel have been promulgated respecting the

arrives . The and sucbetween during variable . structures . These towers are at number , some of them advancing rap others likely to endure for many may here mention one or two peculiarities all . The first is that they stand beside or on tlie site of some ancient burial

does not moment of fluxion though the of time tion merely speaks . The second is , fectly round , tapering upwards from mounted by a hollow overlapping cone after the manner known by the technical rubble "—that is to say , of round stones

one , notcontrary . method of lines , stices of which are smaller stones , or mortar . Conjecture has lost itself assign a date and purpose to these defunct architecture . There is , however investigation may go back . Giraldus

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a very prolix calcudetermination of the because such

deter-, curvilinear spaces , au promotion of geometry the curve , B C the the tangent line = I , the greatest

power ; then y * — — ar , x — ay y ; but if the must be put = O .

; the numerator , quantity iswith

, be of one dimenof three , 3 ; and you x x — a y y . wherein the abcissa x the quantity a ; be of & and then it

, c , x ; then diminish each be 3 a * a ; — 2 x y + therefore—— ayy .

x . ail geometrical curves method , if the scholar mechanical curves be

. the existence throughbuildings , which are towers , " although "baal or beel towers , " of penitence "—all

to the theories that gin of these singular about ninety in y towards decay , but centuries to come . Wo common to them , some ancient church , ground , of which tradiall

They are all built p hrase " sprawled , between the interin to the cement

in endeavouring to strange exemplars of a-, a period from which Cambrcnsis , who lived