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Article FALLACIOUS VIEWS OF THE CRAFT. ← Page 2 of 3 →
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Fallacious Views Of The Craft.
him step by step . In this slig ht sketch a few of his rules are given ; for this glorious branch of science is of immense importance to calculators , and many useful systems have been lost and afterwards revived . Murdoch was well acquainted with Sir Isaac Newton ' s hilosophyas well as that of Leibnitz ; he considered tlie
p , former to be certainly the inventor effluxions ; he maintained that a differential has been , mid still is , by many called a fluxion , ancl a fluxion a differential—yet it is an abuse of terms ; a fluxion has no relation to a differential , nor a differential to a fluxion . The principles upon which the methods are foundedshow them to he . very
differentnot, , withstanding the way of investigation in each be the same , and that both centre in the same conclusion . Nov can the differential method perform what the fltixionary method can . The excellency of the fluxionary method is far above the
diflerential . Mr . Murdoch explained his theory thus : —Magnitudes , as made up of an infinite , number of very small constituent parts put together , are the root of the differential calculus . But by the fluxionary method , wo are taught to consider magnitudes as generated by motion . A described line in this way is not generated by an opposition of pointsor
, differentials , but by the motion or flux of a point : and the velocity of a generating point in the first moment of its formation , or generation , is called a fluxion . In forming magnitudes after the differential way , we conceive them as made up of an infinite number of small constituent parts , so disposed as to produce a magnitude of a given form ; that
these parts are to each other as the magnitudes of ivhich the } ' are differentials ; and that one infinitely small part or differential must be infinitely great , with respect to another diflerential , or infinitely small part ; but by fluxion , or the law of flowing , we determine the proportion of magnitudes , one to another , from the celerities of the motion by which
they are generated . This most certainly is the purest abstracted way of reasoning . Our considering the different degrees of magnitude , as arising from an increasing series of mutations of velocity , is much more , simple and less perplexed than the other way ; and the operations founded on fluxions must be more clear , accurate , and convincing , than those that are founded on the diflerential calculus .
There is a great difference m operations—when quantities are rejected because they really vanish ; and when they are rejected , because they are infinitely small : the hitter method , which is tho differential , must leave the mind in ambiguity and confusion , aud cannot in many cases come up to the truth . It is a very great error , then , to call differentials fluxions ;
and quite wrong to begin with the differential method in order to learn the law , or manner of flowing . Mr . Martin Murdoch's system of teaching ivas this : —He first taught arithmetic , trigonometry , geometry , algebra ; tho two latter branches , first in all their parts and improvements , the methods of series , doctrine of proportions , nature of
logarithms , mechanics , and laws of motion ; from thence he proceeded to the pure doctrine of fluxions , and afc last looked into the differential calculus ; and lie declared it would be lost labour for any person to attempt them who was unacquainted wifch these procognita . When he turned to fluxions , the first thing he did was to
instruct the pupil in the arithmetic of exponents , the nature of powers , and the manner of their generation ; he next went to the doctrine of infinite series , and then to the manner of generating mathematical quantities . This generation of quantities was the first step into fluxions , and he so simply explained the nature of them in this operation , that the scholar was able to form a jmt idea of a first fluxion , though thought by many to be incomprehensible . He proceeded from thence to the notation and algorithm of first
fluxions ; to tho finding second , third , etc ., fluxions ; tlie finding fluxions of exponential quantities , and the fluents from given fluxions ; to their uses in drawing tangents to curves ; in finding the areas of spaces , the values of surfaces , and the contents of solids , their percussion and oscillation , and centre of gravity .
By following his plan , this clever master made the pupil by his explanations happily understand and work with ease ; and made him find no more difficulty in conceiving an adequate notion of a nascent or evanescent quantity , than in forming a true idea of a mathematical point . He gave two years for his pupils to acquire an aptitude to understand
the fundamental principles and operations at all relative to fluxions ; and they could then investigate , and not only give the solution of the most general and useful problems in the mathematics , but likewise solve several problems that occur in the phenomena of nature . The following arc ; some of his difficult questions , which , by
his tuition , were answered immediately : — First question . —He requested in the first place to be informed how the time of a body ' s descending through an arch of a cycloid was found : and if ten hundred weight avoirdupoise , hanging on a bar of steel perfectly elastic and supported at both ends will just break the bar , what must be the weight
of a globe , falling perpendicular 185 feet on tho middle ofthe bar , to liave the same effect '" ! Second question . —How long , and how far , ought a given g lobe to descend by its comparative weight in a medium of a given density , but without resistance , to acquire the greatest velocity it is capable of in descending ivith the same wei ght
, and in the same medium , with resistance : and , how are we to find tho value of a solid formed by the rotation of this curvilinear space , A , C , D . The general equation expressing the nature of the curves : —
m n r , . a — X x x" ) Belli ? V ° yz j TO rt "
Again , how is the centre of gravity to be found of the space enclosed by an hyperbola and its asymp r 7 tote : and , how are we to find the centre of \ ^ y S oscillation of a sphei * e revolving about the j—" 7 f line P , A , M , a tangent , to the generating \ p / \ ^ circle F , A , II , in the point A as an axis ? a
There were some learned men of his time would not allow that a velocity which continues for no time at all can possibly describe any space at all : its effect , they say , is absolutely nothing , and instead of satisfying reason with truth and precision , Iho human faculties are quite confounded , lost , and bewildered in fluxions . A velocity , or fluxion , is at best he does nofc know what—whether something or nothing : and
how can the mind lay hold on , or form any accurate abstract idea of , such a subtile fleeting thing . Mr . Murdoch answered—Disputants may jierplex with dee ]} speculations and confound with mysterious disquisitions , but the method of fluxions has no dependence on such things . The operation is not what any single abstract velocity can
generate , or describe , or assert , but what a continual and successivel y variable velocity can produce in the whole ; and certainly a variable cause may produce a variable effect , as well as a permanent cause a permanent and constant effect ; the difference can only be—that the continual variation of the effect must be proportioned to the continual variation of
the cause . The method of fluxion therefore is true whether he can or cannot conceive the nature and manner of several things relating to them , though he had no idea of perpetually arising increments and magnitudes in nascent or evanescent states . The knowledgeofsuch things is not essential to fluxions
Note: This text has been automatically extracted via Optical Character Recognition (OCR) software.
Fallacious Views Of The Craft.
him step by step . In this slig ht sketch a few of his rules are given ; for this glorious branch of science is of immense importance to calculators , and many useful systems have been lost and afterwards revived . Murdoch was well acquainted with Sir Isaac Newton ' s hilosophyas well as that of Leibnitz ; he considered tlie
p , former to be certainly the inventor effluxions ; he maintained that a differential has been , mid still is , by many called a fluxion , ancl a fluxion a differential—yet it is an abuse of terms ; a fluxion has no relation to a differential , nor a differential to a fluxion . The principles upon which the methods are foundedshow them to he . very
differentnot, , withstanding the way of investigation in each be the same , and that both centre in the same conclusion . Nov can the differential method perform what the fltixionary method can . The excellency of the fluxionary method is far above the
diflerential . Mr . Murdoch explained his theory thus : —Magnitudes , as made up of an infinite , number of very small constituent parts put together , are the root of the differential calculus . But by the fluxionary method , wo are taught to consider magnitudes as generated by motion . A described line in this way is not generated by an opposition of pointsor
, differentials , but by the motion or flux of a point : and the velocity of a generating point in the first moment of its formation , or generation , is called a fluxion . In forming magnitudes after the differential way , we conceive them as made up of an infinite number of small constituent parts , so disposed as to produce a magnitude of a given form ; that
these parts are to each other as the magnitudes of ivhich the } ' are differentials ; and that one infinitely small part or differential must be infinitely great , with respect to another diflerential , or infinitely small part ; but by fluxion , or the law of flowing , we determine the proportion of magnitudes , one to another , from the celerities of the motion by which
they are generated . This most certainly is the purest abstracted way of reasoning . Our considering the different degrees of magnitude , as arising from an increasing series of mutations of velocity , is much more , simple and less perplexed than the other way ; and the operations founded on fluxions must be more clear , accurate , and convincing , than those that are founded on the diflerential calculus .
There is a great difference m operations—when quantities are rejected because they really vanish ; and when they are rejected , because they are infinitely small : the hitter method , which is tho differential , must leave the mind in ambiguity and confusion , aud cannot in many cases come up to the truth . It is a very great error , then , to call differentials fluxions ;
and quite wrong to begin with the differential method in order to learn the law , or manner of flowing . Mr . Martin Murdoch's system of teaching ivas this : —He first taught arithmetic , trigonometry , geometry , algebra ; tho two latter branches , first in all their parts and improvements , the methods of series , doctrine of proportions , nature of
logarithms , mechanics , and laws of motion ; from thence he proceeded to the pure doctrine of fluxions , and afc last looked into the differential calculus ; and lie declared it would be lost labour for any person to attempt them who was unacquainted wifch these procognita . When he turned to fluxions , the first thing he did was to
instruct the pupil in the arithmetic of exponents , the nature of powers , and the manner of their generation ; he next went to the doctrine of infinite series , and then to the manner of generating mathematical quantities . This generation of quantities was the first step into fluxions , and he so simply explained the nature of them in this operation , that the scholar was able to form a jmt idea of a first fluxion , though thought by many to be incomprehensible . He proceeded from thence to the notation and algorithm of first
fluxions ; to tho finding second , third , etc ., fluxions ; tlie finding fluxions of exponential quantities , and the fluents from given fluxions ; to their uses in drawing tangents to curves ; in finding the areas of spaces , the values of surfaces , and the contents of solids , their percussion and oscillation , and centre of gravity .
By following his plan , this clever master made the pupil by his explanations happily understand and work with ease ; and made him find no more difficulty in conceiving an adequate notion of a nascent or evanescent quantity , than in forming a true idea of a mathematical point . He gave two years for his pupils to acquire an aptitude to understand
the fundamental principles and operations at all relative to fluxions ; and they could then investigate , and not only give the solution of the most general and useful problems in the mathematics , but likewise solve several problems that occur in the phenomena of nature . The following arc ; some of his difficult questions , which , by
his tuition , were answered immediately : — First question . —He requested in the first place to be informed how the time of a body ' s descending through an arch of a cycloid was found : and if ten hundred weight avoirdupoise , hanging on a bar of steel perfectly elastic and supported at both ends will just break the bar , what must be the weight
of a globe , falling perpendicular 185 feet on tho middle ofthe bar , to liave the same effect '" ! Second question . —How long , and how far , ought a given g lobe to descend by its comparative weight in a medium of a given density , but without resistance , to acquire the greatest velocity it is capable of in descending ivith the same wei ght
, and in the same medium , with resistance : and , how are we to find tho value of a solid formed by the rotation of this curvilinear space , A , C , D . The general equation expressing the nature of the curves : —
m n r , . a — X x x" ) Belli ? V ° yz j TO rt "
Again , how is the centre of gravity to be found of the space enclosed by an hyperbola and its asymp r 7 tote : and , how are we to find the centre of \ ^ y S oscillation of a sphei * e revolving about the j—" 7 f line P , A , M , a tangent , to the generating \ p / \ ^ circle F , A , II , in the point A as an axis ? a
There were some learned men of his time would not allow that a velocity which continues for no time at all can possibly describe any space at all : its effect , they say , is absolutely nothing , and instead of satisfying reason with truth and precision , Iho human faculties are quite confounded , lost , and bewildered in fluxions . A velocity , or fluxion , is at best he does nofc know what—whether something or nothing : and
how can the mind lay hold on , or form any accurate abstract idea of , such a subtile fleeting thing . Mr . Murdoch answered—Disputants may jierplex with dee ]} speculations and confound with mysterious disquisitions , but the method of fluxions has no dependence on such things . The operation is not what any single abstract velocity can
generate , or describe , or assert , but what a continual and successivel y variable velocity can produce in the whole ; and certainly a variable cause may produce a variable effect , as well as a permanent cause a permanent and constant effect ; the difference can only be—that the continual variation of the effect must be proportioned to the continual variation of
the cause . The method of fluxion therefore is true whether he can or cannot conceive the nature and manner of several things relating to them , though he had no idea of perpetually arising increments and magnitudes in nascent or evanescent states . The knowledgeofsuch things is not essential to fluxions