FOOTNOTES:

FOOTNOTES:[16]We do not mean to imply that theonlyuse made of the Jacquard cards is that of regulating the algebraicaloperations. But we mean to explain that those cards and portions of mechanism which regulate theseoperations, are wholly independent of those which are used for other purposes. M. Menabrea explains that there arethreeclasses of cards used in the engine for three distinct sets of objects, viz.Cards of the Operations,Cards of the Variables, and certainCards of Numbers. (Seepages 13and22.)[17]In fact such an extension as we allude to, would merely constitute a further and more perfected development of any system introduced for making the proper combinations of the signs plus and minus. How ably M. Menabrea has touched on this restricted case is pointed out inNote B.[18]The machine might have been constructed so as to tabulate for a higher value ofnthan seven. Since, however, every unit added to the value ofnincreases the extent of the mechanism requisite, there would on this account be a limit beyond which it could not be practically carried. Seven is sufficiently high for the calculation of all ordinary tables.The fact that, in the Analytical Engine, the same extent of mechanism suffices for the solution ofDelta^{n} u_z = 0,whethern=7,n = 100,000,orn= any number whatever, at once suggests how entirely distinct must be the nnature of the principlesthrough whose application matter has been enabled to become the working agent of abstract mental operations in each of these engines respectively; and it affords an equally obvious presumption, that in the case of the Analytical Engine, not only are those principles in themselves of a higher and more comprehensive description, but also such as must vastly extend thepracticalvalue of the engine whose basis they constitute.[19]This subject is further noticed inNote F.[20]A fuller account of the manner in which thesignsare regulated, is given on in Mons. Menabrea’s Memoir,pages 17,18. He himself expresses doubts (in a note of his own at the bottom of the latter page) as to his having been likely to hit on the precise methods really adopted; his explanation In being merely a conjectural one. That itdoesaccord precisely with the fact is a remarkable circumstance, and affords a convincing proof how completely Mons. Menabrea has been imbued with the true spirit of the invention. Indeed the whole of the above Memoir is a striking production, when we consider that Mons. Menabrea had had but very slight means for obtaining any adequate ideas respecting the Analytical Engine. It requires however a considerable acquaintance with the abstruse and complicated nature of such a subject, in order fully to appreciate the penetration of the writer who could take so just and comprehensive a view of it upon such limited opportunity.[21]This adjustment is done by hand merely.[22]It is convenient to omit the circles whenever the signs + or — can be actually represented.[23]See the diagram ofpage 46.[24]We recommend the reader to trace the successive substitutions backwards from (1.) to (4.), in Mons. Menabrea’s Table. This he will easily do by means of the upper and lower indices, and it is interesting to observe how eachV_7successively ramifies (so to speak) into two otherV′s’s in some other column of the Table; until at length theV′s’s of the original data are arrived at.[25]This division would be managed by ordering the number two to appear on any separate new column which should be conveniently situated for the purpose, and then directing this column (which is in the strictest sense aWorking-Variable) to divide itself successively withV_32,V_33,&c.[26]It should be observed, that were the rest of the factor ({A} + {A} cos theta + &c.)taken into account, instead offourterms only,C_3would have the additional term{1}/{2}{B}_1{A}_4;andC_4,the two additional terms,{B}{A}_4,{1}/{2}{B}_1{A}_5.This would indeed have been the case had evensixterms been multiplied.[27]A cycle that includesnother cycles, successivelycontained one within another, is called a cycle of then+1th order. A cycle may simply include many other cycles, and yet only be of the second order. If a series follows a certain law for a certain number of terms, and then another law for another number of terms, there will be a cycle of operations for every new law; but these cycles will not becontained one within another,—they merelyfollow each other. Therefore their number may be infinite without influencing theorderof a cycle that includes a repetition of such a series.[28]The engine cannot of course compute limits for perfectlysimpleanduncompoundedfunctions, except in this manner. It is obvious that it has no power of representing or of manipulating with any butfiniteincrements or decrements; and consequently that wherever the computation of limits (or of any other functions) depends upon the direct introduction of quantities which either increase or decreaseindefinitely, we are absolutely beyond the sphere of its powers. Its nature and arrangements are remarkably adapted for taking into account allfiniteincrements or decrements (however small or large), and for developing the true and logical modifications of form or value dependent upon differences of this nature. The engine may indeed be considered as including the whole Calculus of Finite Differences; many of whose theorems would be especially and beautifully fitted for development by its processes, and would offer peculiarly interesting considerations. We may mention, as an example, the calculation of the Numbers of Bernoulli by means of theDifferences of Nothing.[29]See the diagram at the end of these Notes.[30]It is interesting to observe, that so complicated a case as this calculation of the Bernoullian Numbers, nevertheless, presents a remarkable simplicity in one respect; viz., that during the processes for the computation ofmillionsof these Numbers, no other arbitrary modification would be requisite in the arrangements, excepting the above simple and uniform provision for causing one of the data periodically to receive the finite increment unity.

[16]We do not mean to imply that theonlyuse made of the Jacquard cards is that of regulating the algebraicaloperations. But we mean to explain that those cards and portions of mechanism which regulate theseoperations, are wholly independent of those which are used for other purposes. M. Menabrea explains that there arethreeclasses of cards used in the engine for three distinct sets of objects, viz.Cards of the Operations,Cards of the Variables, and certainCards of Numbers. (Seepages 13and22.)

[17]In fact such an extension as we allude to, would merely constitute a further and more perfected development of any system introduced for making the proper combinations of the signs plus and minus. How ably M. Menabrea has touched on this restricted case is pointed out inNote B.

[18]The machine might have been constructed so as to tabulate for a higher value ofnthan seven. Since, however, every unit added to the value ofnincreases the extent of the mechanism requisite, there would on this account be a limit beyond which it could not be practically carried. Seven is sufficiently high for the calculation of all ordinary tables.The fact that, in the Analytical Engine, the same extent of mechanism suffices for the solution ofDelta^{n} u_z = 0,whethern=7,n = 100,000,orn= any number whatever, at once suggests how entirely distinct must be the nnature of the principlesthrough whose application matter has been enabled to become the working agent of abstract mental operations in each of these engines respectively; and it affords an equally obvious presumption, that in the case of the Analytical Engine, not only are those principles in themselves of a higher and more comprehensive description, but also such as must vastly extend thepracticalvalue of the engine whose basis they constitute.

The fact that, in the Analytical Engine, the same extent of mechanism suffices for the solution ofDelta^{n} u_z = 0,whethern=7,n = 100,000,orn= any number whatever, at once suggests how entirely distinct must be the nnature of the principlesthrough whose application matter has been enabled to become the working agent of abstract mental operations in each of these engines respectively; and it affords an equally obvious presumption, that in the case of the Analytical Engine, not only are those principles in themselves of a higher and more comprehensive description, but also such as must vastly extend thepracticalvalue of the engine whose basis they constitute.

[19]This subject is further noticed inNote F.

[20]A fuller account of the manner in which thesignsare regulated, is given on in Mons. Menabrea’s Memoir,pages 17,18. He himself expresses doubts (in a note of his own at the bottom of the latter page) as to his having been likely to hit on the precise methods really adopted; his explanation In being merely a conjectural one. That itdoesaccord precisely with the fact is a remarkable circumstance, and affords a convincing proof how completely Mons. Menabrea has been imbued with the true spirit of the invention. Indeed the whole of the above Memoir is a striking production, when we consider that Mons. Menabrea had had but very slight means for obtaining any adequate ideas respecting the Analytical Engine. It requires however a considerable acquaintance with the abstruse and complicated nature of such a subject, in order fully to appreciate the penetration of the writer who could take so just and comprehensive a view of it upon such limited opportunity.

[21]This adjustment is done by hand merely.

[22]It is convenient to omit the circles whenever the signs + or — can be actually represented.

[23]See the diagram ofpage 46.

[24]We recommend the reader to trace the successive substitutions backwards from (1.) to (4.), in Mons. Menabrea’s Table. This he will easily do by means of the upper and lower indices, and it is interesting to observe how eachV_7successively ramifies (so to speak) into two otherV′s’s in some other column of the Table; until at length theV′s’s of the original data are arrived at.

[25]This division would be managed by ordering the number two to appear on any separate new column which should be conveniently situated for the purpose, and then directing this column (which is in the strictest sense aWorking-Variable) to divide itself successively withV_32,V_33,&c.

[26]It should be observed, that were the rest of the factor ({A} + {A} cos theta + &c.)taken into account, instead offourterms only,C_3would have the additional term{1}/{2}{B}_1{A}_4;andC_4,the two additional terms,{B}{A}_4,{1}/{2}{B}_1{A}_5.This would indeed have been the case had evensixterms been multiplied.

[27]A cycle that includesnother cycles, successivelycontained one within another, is called a cycle of then+1th order. A cycle may simply include many other cycles, and yet only be of the second order. If a series follows a certain law for a certain number of terms, and then another law for another number of terms, there will be a cycle of operations for every new law; but these cycles will not becontained one within another,—they merelyfollow each other. Therefore their number may be infinite without influencing theorderof a cycle that includes a repetition of such a series.

[28]The engine cannot of course compute limits for perfectlysimpleanduncompoundedfunctions, except in this manner. It is obvious that it has no power of representing or of manipulating with any butfiniteincrements or decrements; and consequently that wherever the computation of limits (or of any other functions) depends upon the direct introduction of quantities which either increase or decreaseindefinitely, we are absolutely beyond the sphere of its powers. Its nature and arrangements are remarkably adapted for taking into account allfiniteincrements or decrements (however small or large), and for developing the true and logical modifications of form or value dependent upon differences of this nature. The engine may indeed be considered as including the whole Calculus of Finite Differences; many of whose theorems would be especially and beautifully fitted for development by its processes, and would offer peculiarly interesting considerations. We may mention, as an example, the calculation of the Numbers of Bernoulli by means of theDifferences of Nothing.

[29]See the diagram at the end of these Notes.

[30]It is interesting to observe, that so complicated a case as this calculation of the Bernoullian Numbers, nevertheless, presents a remarkable simplicity in one respect; viz., that during the processes for the computation ofmillionsof these Numbers, no other arbitrary modification would be requisite in the arrangements, excepting the above simple and uniform provision for causing one of the data periodically to receive the finite increment unity.

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