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To the Editor of the Monthly Magazine.

SIR,

IME the least comprehensible of

Tevis the least compat appertain to the nature of existence. Logicians have always disagreed in their definitions of it; and our conceptions are little improved respecting it, since the philosophers of Greece puzzled themselves and inankind on this and many other subjects, about two thousand years ago.

These truths, however, appear to be felt, that Time at once generates and devours all things; that it is the medium of existence, or of sensation; that we cannot conceive any mode of existence unconnected with it, and consequently are obliged to admit an Eternity of time past, and of time future.

Yet, however sublime may be the march of Time, as it regards existence in the aggregate, it is impossible not to be sensible of its relative proper ties, as it affects the mind of man. It is evident that we measure it by the combined variety and force of impressions made on the mind; that we have abridged seasons of great vacuity or sameness; and others filled with strong impressions, which double or treble the perceptions of any given period.

No one need be reminded of the length of weeks of adversity; and there are few so radically oppressed by the knavery of the world and of lawyers, as not to have felt the comparative shortness of weeks of pleasure. Every one must also have been sensible of the length of periods, accompanied by change of scene, and novelty of ideas; and of the relative diminution of similar periods passed without variety and care. We are affected in regard to Time as we are by the winds and waves during a sea voyage; if the wind is fair, and the sea unruffled, we go forward a hundred miles without being sensible of our course; but if the wind is stormy, and the sea rolls violently, every mile makes more vivid impressions than would the hundred miles under opposite circumstances. In like manner, if we travel twenty miles on a road with which we are familiar, we receive few or no impressions, and the two ends of our journey as inatter of reminiscence appear to meet; but if we travel over twenty miles of a road we never travelled before, the impressions are numerous, and the apparent distance expands to many times that of the other road, with which we are familiar. Thus it is that unvarying time presents such varying MONTHLY MAG. No. 213.

impressions to the minds of the same men at different periods.

This principle operates also in a simi

lar and uniform manner with reference to the whole progress of the same life. Under similar circumstances, either of sameness or variety, time appears to become shorter as life advances, or as our familiarity with it increases.

Every one who has attained the age of forty must be sensible of the great apparent duration of the early periods of his life, compared with that of the latter periods. The rapid stealth of time is the universal complaint of every one as he advances in age. He feels it, but does not examine, or does not understand its cause. He deplores in vain the rapid passage of weeks, months, years, and decades of years! He remembers the slow and solemn progress of his school days,-how he measured the tardy hours from meal to meal, and from day to day,-how remote was Sunday from Sunday!--Now the day pass. es before he can turn himself;-the year revolves before he can execute any meditated project;-thirty absorbs twenty before he could have supposed it;

he finds himself forty as in a dream; -at fifty he feels himself mocked by the advance of age, and wonders what are become of the last ten years; and at sixty his growing infirmities, by diminishing his enjoyments, and his sources of variety, reduce to a narrow span all that passes in perception of existence, till, by the accelerated motion of time, he is hurried into the grave!

This universal sensation, so intimately blended with our existence and enjoyments, has not, that I know of, been analyzed, or reduced to any practicable view by any writer ancient or modern. Yet surely amidst speculative enquiries, this subject cannot be considered as uninteresting; and although we may not be able to arrest the march of time, or post. pone the period of our dissolution, we may thus be enabled to make a just estimate of our little span of existence; and save ourselves the mortification which may arise from total ignorance of the fleeting nature of our latter days.

The abstract cause of these phenomena regarding time, may be explained in the following manner. We measure nascent or passing time by a nixed feeling arising out of the impressions of the moment, and of the proportion of those impressions to the impressions we have already expe rienced in the time that we have lived.

SK

In

one.

In other words, having no ideas besides
those derived from our experience, we
measure, in general, all future impres-
sions by the number of past ones; and
every given future period is to every
equal past period in the inverse propor
tion of the length of past life. Thus
supposing the powers of reason and re-
tention to commence at five years of age,
the year that passes from seven to eight
will be one half of all past existence,
and will consequently be of great appa-
rent duration; but the year that passes
from twenty to twenty-one will be but
a fifteenth part of all past existence, and
will therefore in its impression on the
mind, be greatly less than the former year.
The consideration, however, is a mixed
If the recollections of all events
were equal, and if events at different
periods were exactly alike, then the ra-
tios of apparent time, at different ages
of the same life, would be as above;
but as recent impressions are so much
stronger than remote ones, and some
events mark a period more emphatically
than others, the ratio is rather to be as-
certained from the experience of mankind,
than from reasoning à priori. Nor can
we reduce so subtle and varied a prin-
ciple to the nice proportion of successive
months or years; but periods of five or
ten years, which average modes of life,
and varieties of impressions, are to be
preferred for such a purpose. Specu-
Jative mathematicians may amuse them-.
selves by drawing out tables calculated
for the smallest periods, but every moral
purpose will be effected by the results of
a general calculation.

"Dividing life then into periods of five years or sixty months; considering the period of infancy as extending to fiveyears; taking one fourth or fifteen months as the proportion arising from proximity, or peculiar force of recent impressions; and taking the successive proportions of sixty, according to the above general principle, the following will be the numbers indicative of the apparent length of every five years in sixty such months as the mind measured in the first five years

of rational existence.

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45 to 50, equal to twenty,
50 to 55 equal to nineteen,
55 to 60 equal to eighteen

After sixty, I conc ive the season of active life is so far gone by, that the ef fect of novelty and variety may be reduced from a fourth to a fifth or sixth; so that at three score and ten, the sixty months of early life will be reduced nearly to an apparent or relative twelve months!

By the table then it appears, that with reference to the apparent duration of the first sixty months of rational existence, the same nominal period will, from the age of twenty to twenty-five be reduced one half; and from forty-five to fifty, will be reduced one third. Hence the five years from twenty to twenty five, will appear, under ordinary circumstances of life, to be only half as long as the pe riod from five to ten, when the mind aequired the greatest stock of sensations and recollections. But the same period will apparently be half as long again as the five years from forty-five to fifty, and twice as long as a siinilar period at sixty!

It appears too that the 660 calendar months which elapse in a man's life between five years of age and sixty, are reduced by this operation of the inind to about one half; so that the apparent and conscious existence which a man has passed at sixty is but the half of its nominal duration! Further, the ten years which elapse between ten and twenty, are equal to the twenty years which elapse from forty to sixty, the two periods in the table being respectively equal to eighty and seventy-eight!

Every man's experience will verify the positions here insisted on, and his feelings will justify the preceding deductions. Others may be made by the contemplative reader, and a variety of strong practical lessons may be inferred at their leisure by moralists and divines. I am content with having called attention' to a principle which I am per. sunded has been felt, without being understood, and which is in all respects too interesting to remain longer among philo sphical desiderata. COMMON SENSE. Buckingham Gate, May 20, 1811.

For the Monthly Magazine.
To w. SAINT, ESQ. of NORWICH.
SIR,

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HOUGH your remarks on my Elements of the True Arithmetic of Infinites, from the asperity with which they are written, and the facility with

which

which they may be answered, demand on my part, that the reply to them should be written with at least, equal severity; yet I will not so far degrade myself in the confutation of them as to act the part of a Reviewer.

In your own language therefore, "Now sir to the point." My three first postu. lates, you say, “You readily grant;" but you are averse to assent to my fourth postulate, which, as you say, runs thus, That to multiply one number, or one series of numbers, by another, is the same thing as to add either of those numbers, or series of numbers, to itself, as often as there are units in the other." You add, "Now to say nothing of the absurdity of calling this a postulate, which is, in reality, a definition, I do not be lieve that it conveys even your own meaning, for surely you will not say that 3, multiplied by 2, is the same as 3 added twice to itself for 3 added once to itself makes 6, and if added twice to itself it

will make 9; and I cannot think, sir, that you meant to say that 3, multiplied by 2, is equal to 9." I have only to say, in answer, that if I am in an error in this instance, your own favourite moderns have, unhappily for me, led me into it. And the first cause of my error was Wolfius, who, in his Algebra, p. 2, says, When unity is contained as oft in one number, as another in a third, the two Dumbers are called factors or co-efficients, and the third is the product, arising from the one drawn into, or mul tiplied by the other, and is no other than adding a number to itself, as often us there are units in the other; but it is done sooner by multiplication." Now that I should be wrong is not at all wonderful, but it seems that even that great modern mathematician Wolfius, is also wrong according to Mr. Saint. perhaps also, sir, you may be of opinion, that a for instance, is not the second power of a, but the first power of it, for a2, you may say is the first multiplication of a by itself. I however, agree with modern mathematicians, that 6 multiplied by 2 is the same thing as adding 6 in itself twice, or 2 to itself six times, and that a is the first power of a, and a the second power of it. You add, "Now if you had to multiply the series, 1+1+1+1 &c. ad infinitum by 1-1, since you have asserted in the corollaries to your first proposition, that 1-1 is that which is neither quantity nor nothing, but which is something belonging to number without being number, you would

And

thus have to add the infinite series 1+1+1+1, &c. to itself as many times as are denoted by that which is neither quantity nor nothing, but which is something belonging to number without being number." Observe sir, with what fa cility this objection may be answered. According to the above citation from Wolfius, the multiplication of two terms is equivalent to the addition of one term to itself, as often as there are units in the other. Now as there are no units in 1-1 it being an infinitesimal, and there are in 1+1+1+, &c. it will be the same thing to add 1-1 to itself, as many times as there are units in the infinite series 1+1+1+1, &c. ns to multiply 1+1+1+1, &c. by 1-1. And so it evidently is according to my theory, For I say, and have demonstrated that 1-1 added to itself infinitely is in the aggre gate equal to 1, though in the distributed form 1-1+1—1+1−1, &c. it is only

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Observe here again, sir, and you will find that your objection vanishes as soon as it is examined. You say, I do not in these instances obtain the remainder +1-1+1-1+1-1, &c. True sir, but what if I obtain a remainder equal to it! Have you any objection to this? Now mark, in every subtraction, if it is truly made, the remainder added to what is subtracted is equal to the subtrahend, by my second postulate, which you say you admit. Consequently sir, to 1-1+1-1+1, &c. let -1+2-1+ 1-1, &c. be added, and the sum is 1; and the like conclusion is true in the other instances. But if this be the case -1+2-1+1-1, &c. is equal to 1-1+ 1-1-1-1, &c. For if to 1-1+1-1 +1-1, &c. 1-1+1-1+1-1, &c. be added, and the sum is 1; and if also to 1-1-1-1+1-1, &c. -1+2—1 +1-1, &c. be added, and the sum is also 1. I think you will not deny, Mr. Saint, that 1-1+1--1, &c. and 1+2 -1+1-1, &c. must be equal to each other. Now, if it clearly appears from all this, that such expressions as 1-1, 1-2+1, &c. are not equivalent to 0, and yet are not quantities, is there any absurdity in asserting that they are ana logous to points at the extremities of lines, which are something belonging to, without being lines; and therefore that these expressions are something belong ing to number, without being number?

Why you exult so much at my having by a very obvious deduction shown the truth of my method of finding the last term of an infinite series, I cannot conceive. For in the eighth proposition, I have demonstrated the truth of this universally, and I chose previously to elucidate it by induction in the third proposition, from the facility with which such induction may be made. My eighth proposition, therefore, is as follows: "In every series of terms in arithmetical or geometrical progression, or in any progression in which the terms mutually exceed each other, the last term is equal to the first term, added to the second term, diminished by the first; added to the third term, diminished by the second; added to the fourth term, diminished by the third; and so on. And if the number of terms be infinite, the last terin equal to the series multiplied by 1-1."

Demonstration: "Iet the terms, whatever the series may be, be represented by a, b, c, d, e, then a+b-a-fc-b+d-c+e-de.

a

+b-a

+d-e +e-d'

e

=

1

-1

But if the number of terms be infinite, viz. if the series be a+b+c+d+e+f+g, &c. ad infin. then this series multiplied by 1-1, will be a+b-a+c-b+d c+e-d+f—e+g—f, &c." Q. E. D. Now, Sir, what becomes of your exul tation; and how came you to be guilty of so unpardonable an omission, as not even to mention this proposition? You have, however, been guilty of a greater and more unpardonable omission than even this. For having granted that the number of terms in an infinite series cannot be greater than and alsa that my method in proposition 3, of obs taining the last term of an infinite series is just; you have wholly neglected to notice the necessary consequence of this concession, which is, the complete subversion of the leading propositions in Dr, Wallis's Arithmetic of Infinites, as I have abundantly shown in the treatise under discussion. Thus in the infinite se ries 0+1+2+3+4,&c. the last or great, est term is 0+1+1+1+1, &c. and the number of terms is 1+1+1+1+1, &c. and 0+1+1+1+1, &c. multiplied by 1+1+1+1, &c. produces 0+1+2+3 +4+35, &c. Thus too in the series 0+1+1+9+16+25, &c.; the last term is 0+1+3+5+7+9+11, &c. and the number of terms is 1+1+1+1, &c. and the last term multiplied by the num her of terms is equal to 0+1+4+9+16, &c. Thus again, in the series 0+1+8 +27+64+125,&c.; the last term is 0+ 1+7+19+37+61, &c. and the number of terms is 1+1+1+1, &c. and the last term multiplied by the number of terms, produces 0+1+8+27+64+125, &c. And so in other instances which are enumerated in prop. 3. Hence, as I infer in corol. 4, to prop. 8. " In every infinite series whether fractional or integral, the terms of which have an uninterrupted continuity, the last term multiplied by the number of terms will be equal to the sum of the series. Now if this, Sir, be admitted to be true, and I defy you, or any mathematician, to show that it is not, the following propositions of Dr. Wallis, are evidently false. "In the arithmetical series 0+1+2+3+4, &c. if the last term be multiplied into the

number

number of terms, the product will be double the sum of all the series."

"In the series of squares 0+1+4+9 +16, &c. infinitely continued, the last term being multiplied into the number of terms, will be triple, to the sum of all the series."

"In the series of cubes 0+1+8+27 +64+125, &c. infinitely continued, the last term being multiplied into the num ber of terms will be quadruple the sum of all the series."

What other reason, Mr. Saint, can be assigned for your omitting to notice this discovery of mine, than a conviction of the truth of it?

You will find your next objection answered in the 29th proposition of my treatise, if you read it with attention; and I shall therefore proceed to your last remark. You ask me "how in taking 1+1 from 2, I obtained the remainder 1-1." It was as follows, Mr. Saint, From Subtract

2

1+1

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Is not the subtraction lawfully made ac cording to the algebraic rule for subtraction? And I also add, is it not ac tually made? For there is no other way of actually subtracting +1 from 0, than by changing the sign. At least this is acknowledged to be the case by all modern writers on algebra. Now, Sir, if the expression 1-1 while it remains in this form, and no actual subtraction is made of 1 from 1, is an infinitesimal, which I have abundantly proved it is, it most clearly follows that 1-1 while it remains in this form, and one unity is not actually added to the other, differs from the aggregate 2 by 1-1.

And now, Sir, I shall conclude with thanking you for the opportunity you have afforded me of vindicating my Arith metic of Infinites, and also for the com pliment you have paid to my heart; but it would have been better, if, in doing it, you had not run your head against mine, as I am afraid it has injured yours. THOMAS TAYLOR. Manor Place, Walworth, May 7th, 1811.

TH

MEMOIRS AND REMAINS OF EMINENT PERSONS.

MEMOIRS OF THE LATE

PAUL SANDBY, Esq. R. A. &c. &c. HOUGH the subject of this Memoir has left behind him that, which will, in time to come, distinguish him from the common dead; a few facts, relating to an individual, whose long career and exertions of eminent talents, have been a public good, will, I presume, be acceptable to the numerous readers of the Monthly Magazine.

Mr. Paul Sandby was born in Not tingham, in the year 1726, and came to Loudon at the early age of sixteen; was soon after placed in the drawing-room in the Tower, (instituted for the purpose of instructing persons in drawing military plans, &c.) and from thence he was selected to attend the survey of the Highlands of Scotland, (as draughtsman) then carrying on under Colonel David Watson, and which took place soon after the rebellion in 1745.

As circumstances are the great governors of men, and may in most instances be said to be the makers of them; perhaps, the destination of Mr. Sandby to the Highlands was the source

of his eminence as a landscape painter, at least in the formation of his pecu. liar style, as, though he there saw nature in her wildest form, the necessity under which he lay of attending to particular accuracy in filling up the plans, may be supposed to have formed in him that correct and faithful habit, with which he after viewed and deli neated her.

It is now too late, (except perhaps from his intimate connexions,) to learn how he passed his early days, or under whose superintendence he received his education; but from the respectable and ancient family from whom he sprung, and his personal and mental acquirements, it was evident that he had been carefully attended to. The cir cumstances that led to his professional excellence are more our iminediate enquiry, and more interesting to others, and to those especially who are to fol low his pursuits in art.

In the life of a painter little variety is to be looked for; the next day being but a repetition of the last, and the succeeding one varying only as he creeps on towards perfection. Mr. Sandby,

however,

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