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Article FALLACIOUS VIEWS OF THE CRAFT. ← Page 3 of 3 Article VOICES FROM RUINS. Page 1 of 3 →
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Fallacious Views Of The Craft.
all they propose is , to determine the velocity or flowing , wherewith a generated quantity increases , ancl to sum up all that has been generated or described by tho continually variable fluxion . On these two bases fluxions stand . Here follow two of Mr . Murdoch's instances : — 1 st . — A heavy body descends perpendicularly , 1 G12 feet
in a second , aucl at the end of this time has acquired a velocity of 32 * 0 feet in a second , which is accurately known ; at any given distance then the body fell , take the point A in the right line , and the velocity of the falling body in the point may be truly computed ; but the velocity in any point above A , at ever so small a distance , will be less than in A ,
and the velocity at any point below A , tit the least possible distance , will be greater than in A . It is therefore plain , that in the point A the body has a certain determined velocity which belongs to no other point in the whole line . Now this velocity is the fluxion of that right line in the point A , and with it the body would proceed , if gravity acted no longer on the body ' s arrival at A .
2 nd . —Take a glass tube open at both ends , whose concavity is of different diameters iu different places , and immerse it iu a stream till the water fills the tube and flows through it ; then in different parts of the tube the velocity of the water will be as the squares of the diameters , and' of consequence different . Suppose then in any marked place a plane to pass through the
tube perpendicular to the axis , or to the motion of the water , and of consequence the water will pass through this section with a certain determined velocity . But if another section be drawn ever so near the former , the water , by reason of the different diameters , will flow through this with a velocity different from what it did at the former ; and therefore to
one section of the tube , or single point onl y , the determinate velocity belongs . It is the fluxion of the space which the fluid describes at that section , and ivith that uniform velocit y the fluid would continue to move , if the diameter was the same to the end of the tube .
3 rd . —If a hollow cylinder bo filled with water , to flow freely out through a hole at the bottom , the velocity of the effluent will bo as the height of the water ; and since the surface of the incumbent fluid descends without stop , the velocity of the stream will decrease , till the effluent be all out . There can then be no two moments of time succeeding
each other over so nearly , wherein the velocity of tlie water is the same ; and of consequence the velocity at any given point belongs onl y fco that particular indivisible moment of time . Now this is accurately the fluxion of the fluid then flowing ; and if , at that instant , more water was poured into the cylinder to make the surface keep its placethe effluent
, would retain its velocity , and still be the fluxion ofthe fluid . Such are the operations of nature , and they visibly confirm the nature of fluxion . It is from hence quite clear thafc the fluxion of a generated quantity cannot retain any one determined value , for the least space of time whatever , but the moment it arrives
at that value , the same moment it loses it again . The fluxion of such quantity can only pass gradually and successivel y through the indefinite degrees , contained between tlie two extreme values , which are the limits thereof during the generation of the fluent , in case the fluxion be variable . But then , though a determinate degree of fluxion does not
continue at all , yet at every determinate indivisible moment of time , every fluent has some determinate degree of fluxion whose abstract value is determinate in itself ; though the fluxion has no determined value for the least space of time whatever . To find its value then , that is , the ratio one fluxion has to anoher , is a problem strictly geometrical ,
notAvithstanding antimathematicians have declared the contrary . Mr . Murdoch ' s was a most ingenious and new method of determinating expeditiousl y the tangents of curved lines ,
which a mathematical reader often finds a very prolix calculus in the common way ; and as the determination of the tangents of curves is of tho greatest use , because such determinations exhibit the gradations of curvilinear spaces , au easy method in doing the thing , is a promotion of geometry in the best manner .
The rule is this : —Suppose B D E the curve , B C the abcissa = x , C D the ordinate == y , AB the tangent line = I , and the nature of the curve be such that the greatest power of y ordinate be ou one side of the equation ; then y * — — ar , x x y \ xyy  « + a ay — a a 4 a , x x — ay y ; but if the greatest power of y be wanting , the terms must be put = O .
Then make a fraction and a numerator ; the numerator , by taking all the terms wherein the known quantity is , with all their signs , and if the known quantity be of one dimension , to prefix unity ; and if two , 2 ; if of three , 3 ; and you will have — 3 a j 2 a a y — 2 a x x , a x x — a y y . The fraction , by assuming the terms wherein the abcissa x
occurs , and retaining the . signs , ancl if the quantity a ; be of one dimension , to prefix unity , as above , & c , and then it will be 3 ar' — 2 x x y + x yy — a a j 2 a x x ; then diminish each of these by x , aud the denominator will be 3 a * a ; — 2 x y + y // — a a f 2 a x . This fraction is equal to ABand
therefore—, , _ — 3 of + 2 aay — 2 act , x f axx — ayy . 3 xx — 2 x y x y y — a a + 2 « x . In this easy way may the tangents of ail geometrical curves be exhibited ; and I add , by the same method , if the scholar be skilful , may the tangents of infinite mechanical curves be determined .
Voices From Ruins.
VOICES FROM RUINS .
MOST people are probably aware ot the existence throughout Ireland of a number of ancient buildings , which are from , their form ordinarily called " round towers , " although the learned have named them variously "baal or beel towers , " " fire towers , " " ivatch towers , " " tower of penitence "—all which names tire referable respectively to the theories that have been promulgated respecting the origin of these singular
structures . These towers are at present about ninety in number , some of them advancing rapidl y towards decay , but others likely to endure for many centuries to come . Wo may here mention one or two peculiarities common to them , all . The first is that they stand beside some ancient church , or on tlie site of some ancient burial groundof which
tradi, tion merely speaks . The second is , that they are all perfectly round , tapering upwards from the base , ancl surmounted by a hollow overlapping cone . They are all built after the manner known by the technical p hrase " sprawled rubble "—that is to say , of round stones , between the interstices of which are smaller stoneshammered in to the cement
, or mortar . Conjecture has lost itself in endeavouring to assign a date and purpose to these strange exemplars of adefunct architecture . There is , however , a period from which investigation may go back . Giraldus Cambrcnsis , who lived
Note: This text has been automatically extracted via Optical Character Recognition (OCR) software.
Fallacious Views Of The Craft.
all they propose is , to determine the velocity or flowing , wherewith a generated quantity increases , ancl to sum up all that has been generated or described by tho continually variable fluxion . On these two bases fluxions stand . Here follow two of Mr . Murdoch's instances : — 1 st . — A heavy body descends perpendicularly , 1 G12 feet
in a second , aucl at the end of this time has acquired a velocity of 32 * 0 feet in a second , which is accurately known ; at any given distance then the body fell , take the point A in the right line , and the velocity of the falling body in the point may be truly computed ; but the velocity in any point above A , at ever so small a distance , will be less than in A ,
and the velocity at any point below A , tit the least possible distance , will be greater than in A . It is therefore plain , that in the point A the body has a certain determined velocity which belongs to no other point in the whole line . Now this velocity is the fluxion of that right line in the point A , and with it the body would proceed , if gravity acted no longer on the body ' s arrival at A .
2 nd . —Take a glass tube open at both ends , whose concavity is of different diameters iu different places , and immerse it iu a stream till the water fills the tube and flows through it ; then in different parts of the tube the velocity of the water will be as the squares of the diameters , and' of consequence different . Suppose then in any marked place a plane to pass through the
tube perpendicular to the axis , or to the motion of the water , and of consequence the water will pass through this section with a certain determined velocity . But if another section be drawn ever so near the former , the water , by reason of the different diameters , will flow through this with a velocity different from what it did at the former ; and therefore to
one section of the tube , or single point onl y , the determinate velocity belongs . It is the fluxion of the space which the fluid describes at that section , and ivith that uniform velocit y the fluid would continue to move , if the diameter was the same to the end of the tube .
3 rd . —If a hollow cylinder bo filled with water , to flow freely out through a hole at the bottom , the velocity of the effluent will bo as the height of the water ; and since the surface of the incumbent fluid descends without stop , the velocity of the stream will decrease , till the effluent be all out . There can then be no two moments of time succeeding
each other over so nearly , wherein the velocity of tlie water is the same ; and of consequence the velocity at any given point belongs onl y fco that particular indivisible moment of time . Now this is accurately the fluxion of the fluid then flowing ; and if , at that instant , more water was poured into the cylinder to make the surface keep its placethe effluent
, would retain its velocity , and still be the fluxion ofthe fluid . Such are the operations of nature , and they visibly confirm the nature of fluxion . It is from hence quite clear thafc the fluxion of a generated quantity cannot retain any one determined value , for the least space of time whatever , but the moment it arrives
at that value , the same moment it loses it again . The fluxion of such quantity can only pass gradually and successivel y through the indefinite degrees , contained between tlie two extreme values , which are the limits thereof during the generation of the fluent , in case the fluxion be variable . But then , though a determinate degree of fluxion does not
continue at all , yet at every determinate indivisible moment of time , every fluent has some determinate degree of fluxion whose abstract value is determinate in itself ; though the fluxion has no determined value for the least space of time whatever . To find its value then , that is , the ratio one fluxion has to anoher , is a problem strictly geometrical ,
notAvithstanding antimathematicians have declared the contrary . Mr . Murdoch ' s was a most ingenious and new method of determinating expeditiousl y the tangents of curved lines ,
which a mathematical reader often finds a very prolix calculus in the common way ; and as the determination of the tangents of curves is of tho greatest use , because such determinations exhibit the gradations of curvilinear spaces , au easy method in doing the thing , is a promotion of geometry in the best manner .
The rule is this : —Suppose B D E the curve , B C the abcissa = x , C D the ordinate == y , AB the tangent line = I , and the nature of the curve be such that the greatest power of y ordinate be ou one side of the equation ; then y * — — ar , x x y \ xyy  « + a ay — a a 4 a , x x — ay y ; but if the greatest power of y be wanting , the terms must be put = O .
Then make a fraction and a numerator ; the numerator , by taking all the terms wherein the known quantity is , with all their signs , and if the known quantity be of one dimension , to prefix unity ; and if two , 2 ; if of three , 3 ; and you will have — 3 a j 2 a a y — 2 a x x , a x x — a y y . The fraction , by assuming the terms wherein the abcissa x
occurs , and retaining the . signs , ancl if the quantity a ; be of one dimension , to prefix unity , as above , & c , and then it will be 3 ar' — 2 x x y + x yy — a a j 2 a x x ; then diminish each of these by x , aud the denominator will be 3 a * a ; — 2 x y + y // — a a f 2 a x . This fraction is equal to ABand
therefore—, , _ — 3 of + 2 aay — 2 act , x f axx — ayy . 3 xx — 2 x y x y y — a a + 2 « x . In this easy way may the tangents of ail geometrical curves be exhibited ; and I add , by the same method , if the scholar be skilful , may the tangents of infinite mechanical curves be determined .
Voices From Ruins.
VOICES FROM RUINS .
MOST people are probably aware ot the existence throughout Ireland of a number of ancient buildings , which are from , their form ordinarily called " round towers , " although the learned have named them variously "baal or beel towers , " " fire towers , " " ivatch towers , " " tower of penitence "—all which names tire referable respectively to the theories that have been promulgated respecting the origin of these singular
structures . These towers are at present about ninety in number , some of them advancing rapidl y towards decay , but others likely to endure for many centuries to come . Wo may here mention one or two peculiarities common to them , all . The first is that they stand beside some ancient church , or on tlie site of some ancient burial groundof which
tradi, tion merely speaks . The second is , that they are all perfectly round , tapering upwards from the base , ancl surmounted by a hollow overlapping cone . They are all built after the manner known by the technical p hrase " sprawled rubble "—that is to say , of round stones , between the interstices of which are smaller stoneshammered in to the cement
, or mortar . Conjecture has lost itself in endeavouring to assign a date and purpose to these strange exemplars of adefunct architecture . There is , however , a period from which investigation may go back . Giraldus Cambrcnsis , who lived