thoughts on man-第69章
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But this is not strictly the case。 Mathematics are like those abstract and imaginary existences about which they are conversant。 They may constitute in themselves； and in the apprehension of an infallible being； a science of certainty。 But they come to us mixed and incorporated with our imperfections。 Our faculties are limited； and we may be easily deceived； as to what it is that we see with transparent and unerring clearness； and what it is that comes to us through a crooked medium； refracting and distorting the rays of primitive truth。 We often seem clear； when in reality the twilight of undistinguishing night has crept fast and far upon us。 In a train of deductions； as in the steps of an arithmetical process； an error may have insinuated itself imperceptibly at a very early stage； rendering all the subsequent steps a wandering farther and farther from the unadulterated truth。 Human mathematics； so to speak； like the length of life； are subject to the doctrine of chances。 Mathematics may be the science of certainty to celestial natures； but not to man。
But； if in the case of pure mathematics； we are exposed to the chances of error and delusion； it is much worse with mixed mathematics。 The moment we step out of the high region of abstraction； and apply ourselves to what we call external nature； we have forfeited that sacred character and immunity； which we seemed entitled to boast； so long as we remained inclosed in the sanctuary of unmingled truth。 As has already been said； we know what passes in the theatre of the mind； but we cannot be said absolutely to know any thing more。 In our speculations upon actual existences we are not only subject to the disadvantages which arise from the limited nature of our faculties； and the errors which may insensibly creep upon us in the process。 We are further exposed to the operation of the unevennesses and irregularities that perpetually occur in external nature； the imperfection of our senses； and of the instruments we construct to assist our observations； and the discrepancy which we frequently detect between the actual nature of the things about us and our impressions respecting them。
This is obvious； whenever we undertake to apply the processes of arithmetic to the realities of life。 Arithmetic； unsubjected to the impulses of passion and the accidents of created nature； holds on its course； but； in the phenomena of the actual world； 〃time and chance happeneth to them all。〃
Thus it is； for example； in the arithmetical and geometrical ratios； set up in political economy by the celebrated Mr。 Malthus。 His numbers will go on smoothly enough； 1； 2； 4； 8； 16； 32； as representing the principle of population among mankind； and 1； 2； 3； 4； 5； 6； the means of subsistence； but restiff and uncomplying nature refuses to conform herself to his dicta。
Dr。 Price has calculated the produce of one penny； put out at the commencement of the Christian era to five per cent。 compound interest； and finds that in the year 1791 it would have increased to a greater sum than would be contained in three hundred millions of earths； all solid gold。 But what has this to do with the world in which we live？ Did ever any one put out his penny to interest in this fashion for eighteen hundred years？ And； if he did； where was the gold to be found； to satisfy his demand？
Morse； in his American Gazetteer； proceeding on the principles of Malthus； tells us that； if the city of New York goes on increasing for a century in a certain ratio； it will by that time contain 5；257；493 inhabitants。 But does any one； for himself or his posterity； expect to see this realised？
Blackstone； in his Commentaries on the Laws of England； has observed that； as every man has two ancestors in the first ascending degree； and four in the second； so in the twentieth degree he has more than a million； and in the fortieth the square of that number； or upwards of a million millions。 This statement therefore would have a greater tendency to prove that mankind in remote ages were numerous； almost beyond the power of figures to represent； than the opposite doctrine of Malthus； that they have a perpetual tendency to such increase as would infallibly bring down the most tremendous calamities on our posterity。
Berkeley； whom I have already referred to on another subject； and who is admitted to be one of our profoundest philosophers； has written a treatise＇48＇ to prove； that the mathematicians； who object to the mysteries supposed to exist in revealed religion； 〃admit much greater mysteries； and even falshoods in science； of which he alleges the doctrine of fluxions as an eminent example＇49＇。〃 He observes； that their conclusions are established by virtue of a twofold error； and that these errors； being in contrary directions； are supposed to compensate each other； the expounders of the doctrine thus arriving at what they call truth； without being able to shew how； or by what means they have arrived at it。
＇48＇ The Analyst。
＇49＇ Life of Berkeley； prefixed to his Works。
It is a memorable and a curious speculation to reflect； upon how slight grounds the doctrine of 〃thousands and thousands of suns； multiplied without end； and ranged all around us； at immense distances from each other； and attended by ten thousand times ten thousand worlds；〃 mentioned in the beginning of this Essay； is built。 It may be all true。 But； true or false； it cannot be without its use to us； carefully to survey the road upon which we are advancing； the pier which human enterprise has dared to throw out into the vast ocean of Cimmerian darkness。 We have constructed a pyramid； which throws into unspeakable contempt the vestiges of ancient Egyptian industry： but it stands upon its apex； it trembles with every breeze； and momentarily threatens to overwhelm in its ruins the fearless undertakers that have set it up。
It gives us a mighty and sublime idea of the nature of man； to think with what composure and confidence a succession of persons of the greatest genius have launched themselves in illimitable space； with what invincible industry they have proceeded； wasting the midnight oil； racking their faculties； and almost wearing their organs to dust； in measuring the distance of Sirius and the other fixed stars； the velocity of light； and 〃the myriads of intelligent beings formed for endless progression in perfection and felicity；〃 that people the numberless worlds of which they discourse。 The illustrious names of Copernicus； Galileo； Gassendi； Kepler； Halley and Newton impress us with awe； and； if the astronomy they have opened before us is a romance； it is at least a romance more seriously and perseveringly handled than any other in the annals of literature。
A vulgar and a plain man would unavoidably ask the astronomers； How came you so familiarly acquainted with the magnitude and qualities of the heavenly bodies； a great portion of which； by your own account； are millions of millions of miles removed from us？ But； I believe； it is not the fashion of the present day to start so rude a question。 I have just turned over an article on Astronomy in the Encyclopaedia Londinensis； consisting of one hundred and thirty…three very closely printed quarto pages； and in no corner of this article is any evidence so much as hinted at。 Is it not enough？ Newton and his compeers have said it。
The whole doctrine of astronomy rests upon trigonometry； a branch of the science of mathematics which teaches us； having two sides and one angle； or two angles and one side； of a triangle given us； to construct the whole。 To apply this principle therefore to the heavenly bodies； it is necessary for us to take two stations； the more remote from each other the better； from which our observations should be made。 For the sake of illustration we will suppose them to be taken at the extremes of the earth's diameter； in other words； nearly eight thousand miles apart from each other； the thing itself having never been realised to that extent。 From each of these stations we will imagine a line to be drawn； terminating in the sun。 Now it seems easy； by means of a quadrant； to find the arch of a circle （in other words； the angle） included between these lines t
