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would be able to make would not be sufficiently §< 5 ctrt ^ te to ehabteliita to discover aiiy of those properties which were not obvious at first sight , yet they m ^ ht , arid would be sufficient to stiglir&stf the Mea' of a perfect cirele ;
for ir-ivimiafcl > drbbvidus , that there was a point soi&ewhere about the middle , which ; tyas nearly equi-distant from all jpiirfe of the outside . ' And if , ttpfca taife ; he should define a perfeet eHrefe tb be < a tjsfure bo ^ dfe * pW
a line , v&tifch lie * called the ctrfenlfcferehtte , aid i * 4 ileii is every wlt ^ re equally distant froni a point within it , called by him its Centre , it is evident that he would now be in possession of a standard tb which-he might Tefer in
any of his subsequent researches ; and from this property of his ideal circle , that all his radii are exactly equal , he might proceed to deduce such other properties as he was able to discover ; always cotopating his conclusions with the definition , and not with such approximations to a circle * as he could
make , or might observe in nature . Now this is the very process which mathematicians have adopted . Their senses , in the first instance , presented a variety of figures to them for examination , most of which were rough
and irregular , though some among them , upon a superficial view , had the appearance of regularity ; yet ^ even those , upon a closer examinaMon , were found to have ajrreat number of small inequalities . The general
appearance , however , of any of the tatter was , by supposing all these smftH inequalities removed , suflScient to suggest to the mind the idea of a perfect figure of its kind ; which perfect figure would evidentl y exist only in the imagination—the aescription of ibid ideal 4
figure is catteda definition of it . A definition Of any geometrical ligur ^ if it be a good one , consists ia the enunciation of somefundteacmtttl property of that figure , from which it « otl ^ r properties may be . deduced ; and which likewise distinguishes it from all' other figUTCiSr ^ : ' ^ * '*¦••* ; ' ' ¦ -- ^•' ** : i : ¦ ¦* : ¦ . ?
The definitions of the various figures being once estftbliflh ^ d , the mathematieian no lofig ^ r has ^ course totally form which i ^ tually existB in mtmtiz but in all iri ^ investigations ^ ifefi ^^ the defiMtfon * ^ i ^^ '&M whicii the d ? fiiitioto ^ was t & ^ n . By
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this means-he is cectsda , that is , he Ka $ * the € vkleri 6 e « ot coitecSouariess that he hais a complete and correct » iea ^ f the ¦ figure whose propemies he is in ^ vestigating ; and if he takes care to have the same evidence for each step of the reasoning which he employs , it is imiuife&l ;' that he will have the ^ mghw est-evidence of thect > rreetnessof Ids
conclusion , which it is possible for man , constituted as he is , ta have . - ' From the above it appears that all mathematicat figures are ideal , or exist only in the imagination ; hence the m ^ thenlatician has a complete < ioncep- ^ tion of the figure whose properties he is investigating- —it is a creation of his
own , - arid he has the evidence of consciousness that no circumstance respecting it , however trivial , can escape his notice—he has likewise the same evidence ftyfe every step of his reasoning ; for in every transition which he makes from one property to another ^ he has the immediate evidence of
consciousness whether they agree or disagree , his mind taking cognizance of both at the same instant * Here > then ; are the circumstances which give such-peculiar force to
mathematical evidence or demonstration ; we * know , by conisciousness , that the ; things themselves are completely com- » prehended ; we have the same evidence for every successive step in the demonstration , and at the conclusion
we are cons ^ iou « that we remember this ; but even supposing there should be some part of the demonstration of which we have not a clear and dis ^ tinet remembrance , we have the power of going ovei- the whole again , and of repeating this re ^ exanaination till we are -conscious that we ^ do remember
every part distinctly , till we j ^ re able to ^ make the whole rpaBs in rapid review before the mindV It ia therefore clear that we ha ^ ve the evidence of memory and - consciousness m for - ih& truth of . the conclusion . - But this i »
th « highest kind iojf evidence which it is possible Jfor naanuto have ; it not only doesy : but must abvays , carry ^ irresistible coavic ^ on 4 otliei raUid ^ so long < as ^ th « imiuili of : man has , eju&t eUlMS- ^ - ^ H'imhMU . ^^' - rii ^^; f > lA * ' . tKfi » truthsv
> 4 ; h ^ < O&i ^ ti ^ yf ; ical ' , If jectsds fo ^^ dewe ^ tWt them i ^ ^ o
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Ensay on Truth , 216
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Citation
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Monthly Repository (1806-1838) and Unitarian Chronicle (1832-1833), April 2, 1823, page 215, in the Nineteenth-Century Serials Edition (2008; 2018) ncse.ac.uk/periodicals/mruc/issues/vm2-ncseproduct1783/page/23/
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