Lectures on the
PHILLOSOPHY OF ARITHMETIC.
by URIAH PARKE, 1850.

LECTURE XXII. ARITHMETICAL PRODIGIES, &c.

Having completed our course of investigation into the philosophy of numbers, we shall devote the present lecture to an investigation of the human mind as adapted to the study of this science. In making this announcement, however, we desire not to excite anticipations that are not to be realized; for we have neither space nor inclination to enter into a discussion of the vexed questions of metaphysicians, as to whether the mind is material or immaterial; and how far the external configuration of the head is indicative of the powers of the mind within. It is sufficient for our purpose to advance the fact, that the powers of the human mind, various in all things, seem in this peculiarly unequal, for while the mass of our race require the aid of long continued education to train the mind to the perception of numerical relations, and some seem incapable of reaching a high degree of proficiency, others possess a power, even in untaught childhood, that cannot be reached by the ablest mathematicians.

Some have supposed that mathematical skill, with which they identify the cases referred to, implies a high development of the reasoning faculties; but though the exercise of the mind in the study of mathematical science, has been always admitted to be an excellent mode of discipline, it by no means follows as a legitimate consequence, that the possession of extraordinary perceptive faculties in regard to the powers and relations of numbers, or of quantities, implies extraordinary reasoning faculties. An astonishing degree of perceptive power, in regard to numbers, has been found indeed to exist in minds but slightly removed from downright idiocy: a fact that would seem unaccountable, if the elementary combination of numbers required the aid of the reasoning powers.

A most striking instance of mental imbecility, combined with a high degree of power in regard to numbers, was brought to light in 1844, in the person of a negro slave, named Cap, the property of Mr. P. M'Lemore, of Madison county, Alabama. We cannot present this case more clearly than by giving the following somewhat extensive extract from a letter, dated October 26th, 1844, written by the Rev. John M. Hanner, and subsequently confirmed by the same gentleman, in a letter written in reply to one from us.

"On the 8th of June, 1844, the Rev. John C. Burruss, Mr. T. Brandon and myself went to see him and were amazed. From himself and Mr. M'Lemore, we learned that he has no idea of a God. When asked, " Who made you?" he answered "Nobody." He has never been but a few times half a mile from the place of his birth. He has not mind enough to do the ordinary work of a slave; eats and sleeps in the same house with the white folks, having his own table and bed. He will not ask for any thing, nor touch food, however hungry, unless it be offered to him. He was never known to commence a conversation with any one, nor continue one, farther than merely answering questions in the fewest words. He speaks very low and tardily. He has never been known to utter a falsehood or to steal, and is but little subject to anger. He will not strike, even a dog, but when vexed by his sister, he will take hold of her arm as if to break it with his hands. He cannot be persuaded to taste intoxicating liquors; and manifests no partiality for females. There is nothing remarkable in the configuration of his head or in his countenance, save that his eye is uncommonly convex, and continually rolling about with a wild and glaring expression. His laugh and movements are perfectly idiotic. He does not know a letter or figure. Withal he is in one respect the most extraordinary human being I ever saw. Almost the only manifestation of mind is in relation to Numbers. His power over numbers is at once extraordinary and incredible. Take any two numbers under 100, and he will give their products at once, as readily as a school boy would give the product of 12 times 12. He multiplies thousands, adds, subtracts and divides, with the same certainty, though with greater mental labor. He has, however, no idea of numbers above the period of millions.

With pencil and paper we made the following calculations, and asked him the questions; thus—

How much is 99 times 99? He answered immediately 9,801. How much is 74 times 861. He answered 6401. How many nines in 2000? He answered, 222 nines and 2 over. How many fifteens in 3355? He answered, 223 fifteens and 10 over. How many twenty-threes in 4000? He answered, 173 twenty-threes, and 21 over. How much is 321 times 789? He answered after a short pause, 253,269. If you take 21 from 85, how many will be left? He answered, 64. How much is 7 times 9, 22 and 14? He answered, 99. How much is 17 times 17 and 16? He said, 305. If you had given one dollar and a half for a chicken and a half, how much would you have to give for two chickens? He said, Two Dollars. If a stake three feet long, standing upright, makes a shadow of five feet; how high will a pole be that makes a shadow of thirty feet? At this he put his hand to his chin, drew himself up, and gave a silly laugh. His master said he did not understand such questions as that.

We then asked him, How much is 3333 times 5555? In this instance, as in some of the others, he looked serious, began to twist about in his chair, to pick his clothes and finger nails, to look at his hands, put the points of his thumbs to his teeth, move his lips a little; and then he seemed to think a little, when his countenance gave signs of mental agony, and thus these symptoms continued.

His master told him to walk about and rest himself. He went into the yard, and appeared to be alternately elated with rapture, and depressed with gloom. He would run, jump up, throw his arms into the air above his head; then stand still, and then drag his foot over the weeds, look up and down; in a word he made all sorts of crazy motions. When we rose from the dining table, we found him on the piazza, sitting perfectly composed. He then stated when asked, the amount to be 18,514,815.

We could get no clue to the mental process by which he ascertained such results. When asked how he did it, his unvarying answer was, "I studies it up." But what do you do first and what next? He merely drawled out "I studies it up." He did not count his fingers, nor any thing external, nor did he seem to count at all; and yet he combined thousands and millions, and played with their combinations, just as others would with units. All the instruction he ever received was from his master, who taught him to count 100, and would amuse himself by asking simple questions, such as the twenties, or the fives, in a hundred."

Mr. Hanner saw him a few days afterwards, and found he perfectly recollected the numbers that had been given him on the former occasion; as well as his own answers.

We have since conversed with persons who have seen the above negro, and find Mr. Hanner's account fully confirmed. We might give numerous other answers equally wonderful for one of so little intellect, but we desire not to consume too much space; though we wish to give such description as will enable the mental philosopher to understand the case. In body, Cap weighs nearly 200 pounds, and all agree as to his idiocy. A person who saw him in 1845 says, "Though only 19, he has the appearance of being 30. He does not know a letter or figure, or any other representative of numbers or ideas. He speaks to'no one, except when spoken to. His forehead is low, and covered with hair, within an inch and a half above his eyebrows. But the volume from temple to temple is great beyond comparison. I noticed that even numbers were more easily solved by him than odd ones, but could find no clue to his mode of solution. Such is the Alabama Negro, the wonderful being of one idea!" Had Mr. Hanner been a phrenologist the shape of the forehead would not have passed unnoticed.

Though other instances of mental imbecility in such calculators, have been found, the above seems to be far the most remarkable. There was a white man, living near Metuchin in New Jersey, some years ago; for whom a guardian to take care of his property was necessary, though himself the wonder of his acquaintance, for his powers of calculation. We have been unable to learn the particulars with sufficient accuracy for publication. Fowler, in his Practical Phrenology, speaks of meeting with a case in 1837, at Fairhaven, Massachusetts, in which the calculating power was combined with a great degree of imbecility; but we have been unable to learn reliable particulars.

The state of Vermont has furnished two of the most remarkable cases on record; Zerah Colburn, and Truman H. Safford. The former died at the age of 35, the latter is now at Cambridge, Massachusetts, receiving the full benefit of a collegiate education. As these cases differ from each other as well as from those we have before alluded to, we shall give a pretty full account of both; and we hope the reader will bear in mind the points of resemblance and difference.

Zerah Colburn, was born at Cabot, in the state of Vermont, September 1, 1804; and it is said in his memoirs that of' the seven brothers and sisters who formed the family, he was regarded in infancy as the dullest in intellect. The first exposition of the peculiar powers of his mind in combining numbers, was in August, 1810, when he was about a month under six years of age. While his father was at work at a joiner's bench, Zerah was playing amongst the chips, when his father was surprised to hear him saying to himself, "5 times 7 are 35, 6 times 8 are 48," &c., evidently amusing himself by the process of calculation. He had then been at the district school about six weeks, and his father supposed he might have caught these expressions there; but on examination he found him perfect in the numbers of the common multiplication table, and on proposing other numbers he was found equally accurate. He inquired the product of 97 by 13, when Zerah promptly answered 1261. News of his wonderful powers soon spread through the neighborhood, and many called upon him to satisfy their reasonable incredulity, who going away more than satisfied, spread the tale of wonder, with additions of what they had themselves seen. The account soon found its way into the public papers, and was spread throughout Europe and America. The boy was taken to the seat of government of his native state, where his powers were more fully tried.

Questions in multiplication of two or three places of figures, were answered with much greater rapidity than they could be solved on paper. Questions involving an application of this rule, as in Reduction, Rule of Three, and Practice, seemed to be perfectly adapted to his mind. The Extraction of the Roots of exact Squares and Cubes was done with very little effort; and what has been considered by the Mathematicians of Europe, an operation for which no rule existed, viz; finding the factors of numbers, was performed by him; and in the course of time, he was able to point out his method of obtaining them. Questions in Addition, Subtraction, and Division, were done with less facility, on account of the more complicated and continued effort of the memory. In regard to the higher branches of Arithmetic, he had no rules peculiar to himself; but if the common process was pointed out as laid down in the books, he could carry on this process very readily in his head.

That such calculations should be made by the power of mind alone, even in a person of mature age, and who had disciplined himself by opportunity and study, would be surprising, because far exceeding the common attainments of mankind;—that they should be made by a child six years old, unable to read, and ignorant of the name or properties of one figure traced on paper, without any previous effort to train him to such a task, will not diminish the surprise.

The project of educating him thoroughly was very early suggested, and many propositions were made to his father who traveled with the boy, but though anxious to effect the same object, he seems to have been of an unhappy, suspicious disposition, always fearful of being defrauded or imposed upon; and hence though he traveled with his son through the United States and Europe, and many efforts were made in both countries to aid him, he succeeded but partially in effecting his object; indeed it would have been infinitely better for the son, had he been alone, for the waywardness of the father kept both poor, and prevented the friends of science from effecting their wishes in the profound education of the youth. Mr. Colburn and son embarked for England, April 3, 1812, and took up their residence in London, where Zerah was visited by thousands, among whom were many of the first men of the kingdom. Some who saw him engaged in calculation, speak of his agitation, comparing it to St. Vitus' dance. The following extract from his Memoir, page 37, may show the kind of exercise to which his mind was subjected:

"Among other questions, the Duke of York asked the number of seconds in the time elapsed since the commencement of the Christian Era, 1813 years, 7 months, 27 days. The answer was correctly given: 57,234,384,000. At a meeting of his friends which was held for the purpose of concerting the best method of promoting the interest of the child by an education suited to his turn of mind, he undertook and succeeded in raising the number 8 to the sixteenth power, and gave the answer correctly in the last result, viz; 281,474,976,710,656. He was then tried as to other numbers, consisting of one figure, all of which he raised as high as the tenth power, with so much facility and despatch that the person appointed to take down the results was obliged to enjoin him not to be too rapid. With respect to numbers consisting of two figures, he would raise some of them to the sixth, seventh and eighth power, but not always with equal facility: for the larger the products became, the more difficult he found it to proceed. He was asked the square root of 106,929, and before the number could be written down, he immediately answered 327. He was then requested to name the cube root of 268,336,125, and with equal facility and promptness he replied 625.

Various other questions of a similar nature respecting the roots and powers of very high numbers, were proposed by several of the gentlemen present; to all of which satisfactory answers were given. One of the party requested him to name the factors which produced the number 247,483, which he did by mentioning 941 and 263, which indeed are the only two factors that will produce it. Another of them proposed 171,395, and he named the following factors as the only ones, viz:

5x34279, 7x24485, 59x2905, 83x2065, 35x4897, 295x581, 413x415.

He was then asked to give the factors of 36,083, but he immediately replied that it had none; which in fact was the case, as 36,083 is a prime number."

"It had been asserted and maintained by the French mathematicians that 4294967297 (=2³² + 1) was a prime number; but the celebrated Euler detected the error by discovering that it was equal to 641x6,700,417. The same number was proposed to this child, who found out the factors by the mere operation of his mind. On another occasion, hswas requested to give the square of 999,999; he said he could not do this, but he accomplished it by multiplying 37037 by itself, and that product twice by 27. Ans 999,998,000,001. He then said he could multiply that by 49, which he did: Ans 48,999, 902,000,049. He again undertook to multiply this number by 49: Ans. 2,400,995,198,002,401. And lastly he multiplied this great sum by 25, giving as the final product, 60,024,879,950,060,025. Various efforts were made by the friends of the boy to elicit a disclosure of the methods by which he performed his calculations, but for nearly three years he was unable to satisfy their inquiries. There was, through practice, an increase in his power of computation; when first beginning, he went no farther in. multiplying than three places of figures; it afterwards became a common thing with him to multiply four places by four; in some instances five figures by five have been given.

Some persons had very strange ideas as to the manner in which he reckoned; on one occasion, a gentleman came in, and after putting some questions, began to believe that the boy was assisted by some note or hint furnished to him by some one concealed in the room; he doubted so far as actually to request leave to carry him out into the street at a distance from the house, away from his father, to ascertain whether the same readiness of reply would be evinced.

At another time a man came in while the room was full of company, having something wrapped up in a handkerchief under his arm, and taking the father aside, requested leave to propose as his question, "What book had he in his handkerchief?" he manifested considerable dissatisfaction because the question was not allowed."

By this time the child, then between 8 and 9 years old, had at intervals learned to read and write, and he remarks that he was fond of reading as a pastime.

"In the studies to which he subsequently gave his attention, he manifested no uncommon skill or quickness, though his progress was always respectable. The acquirement of language was easy and pleasant; Arithmetic, (in the books,) entertaining; Geometry, plain but dull."—Mem. p. 40.

In March 1814, a private instructor was employed to teach the youth Mathematics.

"As might be expected from the nature of his early gift, he ever had a taste for figures. To answer questions by the mere operation of mind, though perfectly easy, was not anything in which he ever took satisfaction; for, unless when questioned, his attention was not engrossed by it at all. The study of Arithmetic was not particularly easy to him, but it afforded a very pleasing employment, and even now, were he in a situation to feel justified in such a course, he should be gratified to spend his time in pursuits of this nature. The faculty which he possessed, as it increased and strengthened by practice, so by giving up exhibition, began speedily to depreciate. This was not as some have supposed, on account of being engaged in study; it is more probable to him that the study of any branch that included the use and practice of figures would have served to keep up the facility and readiness of his mind. The study of Algebra, while he attended to it, was very pleasant but when just entering upon the more abstruse rules of the first part, he was taken away from his books and carried to France."—Mem. p. 68.

In 1814, Mr. Colburn and son went to France where the son was fortunate enough to obtain a place in the Lyceum Napoleon, where he spent several months. In 1816 he returned to London, when he obtained a situation at Westminster school where he remained about three years.

In allusion to the study of Geometry, Mr. Colburn remarks:

"Many have inquired if the study of Geometry was easy to him? He never found, that he recollects, any difficulty in understanding the demonstrations laid down by Euclid. Their fitness and adaptation to the various problems or theorems were very evident to his mind, but the study was always dry and devoid of interest. The reason probably was, while studying he did not realize, even in anticipation, the benefits of such a science; had he been engaged in some pursuit that would have required the continual introduction and application of Geometrical principles, the subject would have assumed an interesting appearance, his mind would have been engaged in it, and he would have remembered the principles and arguments laid down."—Mem. p. 114.

After leaving Westminster, Mr. Colburn and his son being both without means of support, the stage was suggested and Zerah was placed under the instruction of Charles Kemble; but his dramatic career was brief, and not flattering; and he afterwards spent two or three years in misery through want of means and want of employment. He was then from 15 to 17 years of age and had long ceased to exhibit his peculiar powers for a support. Indeed his extraordinary powers seemed to leave him, as he acquired general education, and ceased to exercise them.

In 1821 he obtained a school and spent several months in that employment, and in aiding Dr. Young, in astronomical calculations. February 14, 1824, his father died, and soon afterwards the son embarked for America. After visiting home he engaged in teaching and afterwards became a Methodist travelling preacher, and continued in that profession until his death; which occurred in 1839. His remains now lie in the town of Norwich, Connecticut, without a stone to mark the spot. Such is fame. In heart, he was one of the excellent of the earth.

What might have been effected by the aid of a profound education, cannot be known; but it is fair to presume, from his own account of himself, that though he possessed respectable talents, his mental endowments were not of a superior order; neither did he seem to possess common tact for acquiring a livelihood, and the father appears to have had less than the son; hence they were harassed with want, and instead of helping themselves spent their time in soliciting aid from others. The education of Zerah was respectable, although not what his friends desired. He possessed very considerable mathematical knowledge, and was familiar with the French language, which he learned during his stay in France where he also made some proficiency in the German; and during his attendance at Westminster he must have made very considerable proficiency in the Latin: added to all this was the intercourse he necessarily had with mankind during his travels, which was well adapted to improve his mind, for the individuals with whom he associated were generally of the right kind to induce improvement.

The following are some of the methods of calculation pursued by Colburn, as explained in his memoirs.

"In extracting the square root, his first object was to ascertain what number squared would give a sum ending with the last two figures of the given square, and then what number squared will come nearest under the first figure in the given square when it consists of five places. If there are six figures in the proposed sum, the nearest square under the two first figures must be sought, which figures combined will give the sum required;" and the cube root is found by an application of the same principle. The manner in which he proceeded to find the factors of numbers was somewhat similar. He ascertained or rather bore in mind what numbers had certain terminations, and narrowed down his search by the application of established principles bearing upon the case. If for instance the given number was odd, he knew that the factors must be odd, and if it was not divisible by any proposed number, it could not be by any multiple of such number; thus his process was narrowed down, but still left too wide for the skill of the ordinary mind.

His process of multiplying involved less difficulty, and was something like this: he first divided the factors into their round numbers, thus: Multiply 1675 by 325.

1000 600 70 5

300 20 5

Then in his mind he multiplied 1000 by 300 and remembered the product	300,000
Then 600 by 300, and the product 180,000, added to the other, makes	480,000
Then 70 was multiplied by 300, making 21,000, and being added to 480,000 made	501,000
To which lastly the product of 5 by 300 being added we have	502,500
This disposes of the 300, and we take 20 times 1000=20000, which makes	522,500
Then 20 times 600=12000, which added makes	534,500
And 20 times 70=1400, which added makes	535,900
Then 20 times 5=100, which added makes	536,000
This is the product by 320, to which we add the products of the several parts by 5, viz: 1000 by 5=5000, making	541,000
Then 600 by 5=3000, making	544,000
Then 70 by 5=350, making	544,350
Then 5 by 5=25, making the whole product	544,375

This process is peculiar in beginning at the highest place instead of the lowest; but it is plain that for mental operation this is far better, as the large numbers are so much more easily remembered from having no low places until almost the last. It is true that the process is prolix, but it is nature's process, and probably was used before our more artificial mode, and it may be profitable to compare it with the common mode, that both may be better understood; for when they are carefully compared, they will be found very much alike. Let us compare the calculations:

1675 325 8375 3350   5025     Product as before 544,375

1675      325 300000 180000 21000 1500 20000 12000 1400 100 5000 3000 350         25 544,375

He began by multiplying from left to right, so that his products would stand thus, and a little observation will show that it is in effect the same process we daily use, only that we abridge it by carrying as we proceed from right to left. Compare for instance the products by 5, the last four products in the operation, with the product in a single line, and it will be found substantially the same.

Although the foregoing were given by Colburn as his modes of calculation, we are inclined to doubt their accuracy; for some of them seem to pre-suppose a knowledge of figures, which he certainly did not at that time possess. We are rather inclined to the belief that having become, at the time he defined his modes, somewhat acquainted with the use of figures, and being anxious to satisfy the reasonable curiosity of the world to learn his modes of calculation, he deceived himself in using characters while describing a purely mental operation; for as he knew nothing of any representatives of numbers, he must have contemplated numbers themselves.

Perhaps the present will be as favorable a time as any other, to draw a distinction absolutely necessary to be made, between mental calculation by means of figures, and mental calculations without their aid. The latter is what we would understand by the term Mental Arithmetic; bit the term is generally applied to all calculations in which neither sensible objects nor figures are used. Pestallozi carried this practice with his pupils to a very great extent; and every child that commences oral exercises before using characters, must study in the same way; but after learning in the usual way, our mental calculations are very similar to our written ones, and this without reference to the question whether we have studied the subject analytically or synthetically. Perhaps it is almost impossible, after becoming familiar with figures as the representatives of numbers, to calculate numbers entirely in the abstract. Our calculations seem naturally to flow into the common form, only carrying on the operation by concentrating the attention and imagining how the quantities would appear if written. Practice and effort will discipline the mind so as to enable it to produce astonishing results; and yet there may be little invention, and nothing whatever peculiar in the mind.

It is said of the celebrated mathematician, Euler, that two of his pupils having differed in the result of a converging series of seventeen terms, at the fiftieth figure of the result, he reviewed their work mentally, and pointed out the proper correction. This was probably in the latter part of his life, as the loss of his sight then compelled him to cultivate mental calculation, and to avail himself of the aid of an amanuensis. He was then able mentally to raise any number less than a hundred to the fifth or sixth power, without difficulty. He had always however cultivated in some degree the habit of mental calculation.

Dr. Waliston tells us that he himself could in the dark perform multiplication, division, and the extraction of roots to forty decimal places; that he once proposed to himself, while in bed, a number of fifty-three places, and found the square root though extending to twenty-seven figures; and that wNithout writing a single figure, he dictated the result from memory twenty days after.

We sometimes meet with clerks who are able to extend the amounts in bills of items, and to sum up the total, with the apparent rapidity of thought. This may be in part the result of natural quickness, but it is much more dependent on practice and close attention in observing the relations of numbers. For this purpose too it is desirable to be familiar with the products of numbers as far as 20 or 30 at least, instead of 12, the ordinary limit of the multiplication table; and to become familiar with every time and labor saving expedient. The following calculations are said to have been performed by Abraham Hagarman, of Brighton, Monroe County, New York, and though they indicate nothing of the peculiar genius of Cap or of Colburn, they show very clearly the power of concentrated attention and long continued practice; for it is said that mathematical studies, and especially the solution of difficult problems, has occupied his chief attention for thirty years, fourteen of which he has been an invalid. The experiment of mental calculation however has been commenced within a few years. We extract the following from his calculations.

1st. 987654 x 345678 = 341,410,259,412.

2d. 9753214 x 2345678 = 22,877,899,509,092.

3d. 46375619 x 54625125 = 2,533,273,984,827,375.

4th. 123456789 x 123456789 =15,241,578,750,190,521.

5th. 9615324516 x 4256484144 = 40,927,476,341,768,474,304.

6th. 82527613529 x 49243126216 = 4,063,917,689,313,816,176,264.

7th. 951427523675 x 484324256144 = 460,799,427,678,822,324,209,200.

8th. 831532463519 x 643234375246 = 534,870,264,668,411,251,650,674.

9th. 648728416968 x 421875625125 = 273,682,706,444,726,657,121,000.

The first, second, third and fourth of the above operations he accomplished in from one and a half to two hours. The fifth, sixth, seventh, and eighth, occupied from two to three hours. The ninth he accomplished in less than one hour, owing to the favorable character of the multiplier. This is certainly a great feat to be performed "in the head" alone; and shows very clearly what can be done by persevering effort, with perhaps no peculiarity of mental constitution, except a fondness for such amusements. Close attention is all important, it is the great constituent of inventive powers. Sir Isaac Newton says, "It is that complete retirement of the mind within itself, during which the senses are locked up—that intense meditation on which no extraneous idea can intrude—that firm, straight forward progress of thought, deviating into no irregular sally, which can alone place mathematical objects in a light sufficiently strong to illuminate them fully, and preserve the perceptions of the mind's eye in the same order that it moves along."

This power over the attention may be acquired to a great extent, by any one of sound mind, but with very different degrees of readiness, and probably not always to the same extent by different persons. In some this power seems natural, while with others the acquisition costs great labor. The perceptive faculties are very different in different individuals, and this is true in regard to perceiving the relation of numbers, as well as all other mental perceptions. In some, this faculty seems peculiarly obtuse, and they practice calculations with great difficulty. Some men even of fine minds require great effort in order to learn the simplest rules of arithmetic; and Humboldt speaks of the Chaymas, (a people in the Spanish parts of South America,) that have great difficulty in comprehending any thing that belongs to numerical relations; and that the more intelligent count in Spanish, with an air that denotes a great effort of the mind, so far as 30, or perhaps 50. He mentions, as a peculiarity, that the corners of their eyes are turned up towards their temples.

James Garry, who was remarkable for his powers of calculation, resided some years ago at Harper's Ferry, Va., and from J. A. Fitzsimmons, Esq., who was intimate with him, we have obtained the following account.

Mr. Garry was born in the county of Antrim, in Ireland, but immigrated to this country in early life. He was first employed in New York city, at a large salary; and subsequently by Tiffany, Shaw & Co. of Baltimore. While there it was customary for one of the clerks to call over the items of the largest bills of goods, and as rapidly as the clerk could write them down, Garry would give the extension of each line and the footing of the bill; without requiring the clerk to delay a moment, and with absolute certainty of being right. He was subsequently employed as a clerk by Messrs Wager & O'Byrne, of Harper's Ferry, Va., Commission Merchants, where we first heard of him; and where Mr. Fitzsimmons was a fellow clerk with him. In a social point of view, he speaks of him as exceedingly warm hearted, though with a tinge of melancholy, that was probably increased by an unfortunate habit of intemperance. An estimate may be formed of the extent of this singular gift, from the fact that while at Harper's Ferry, his former employers at Baltimore offered him $2000 dollars per annum, if he would return and bind himself to be temperate. But he declined. He was not prepossessing in his manners, and though a tolerable penman, was entirely unacquainted with Grammar, Geography, History &c., his great forte being mental calculation; if that can be called calculation, which seemed to be mere perception. Mr. Fitzsimmons says " His powers of calculation were indeed wonderful, and the gift was natural—not acquired. He was never known to make a mistake, except when working with pen or pencil to show work; and then but seldom.

He could give the sum total of any sum of figures, momentarily, without his ever having been found in error in any case; but he had very little ability for any science except figures. He was almost as prompt in the higher branches of arithmetic as in the elementary; though complex operations evidently cost him thought. He could give no account of his modes of operation, but said the answer came instantly, and stood right before his eyes, and he had only to read what he mentally saw. He said it seemed to be there as by magic. In speaking of the extension and summing up of a long bill of items, he remarked to a friend that "The items seemed to pass before him like the ghosts in Macbeth, at the same time adding themselves together as they overtook each other in the journey; thus increasing in bulk until the whole were united, and the sum total was at once before him."

In answer to the question whether there seemed to be any process of reasoning, Mr. Fitzsimmons stated that the result seemed to be matter of instantaneous perception, and that Mr. Garry so described it; but stated that in difficult problems there were some little delay, and indications of mental effort, but Mr. Garry seemed to think, as he expressed it, that the operation was the same, "only the ghosts rose a little slower, and moved more solemnly." When intoxicated, his answers were rather less prompt, but still accurate.

About 1837 or'38, he visited St. Louis where he died, aged about 38 years.

Jedediah Buxton, of England was another instance. He was uneducated, and wrought his solutions by his native ingenuity. The following is given as one of his performances. "On being required to multiply 456 by 378, he gave the product in a very short time; and when requested to work the question audibly, so that his process might be known, he multiplied 456 first by 5, which produced 2280; this he again multiplied by 20, and found the product 45,600, which was the multiplicand multiplied by 100; this product he again multiplied by 3, which produced 136,800, the product of tre multiplicand by 300. It remained then to multiply by 78, which he effected by multiplying 2280 (the product of the multiplicand by 5) by 15, as 5 times 15 are 75. This product being 34,200, he added to 136,800, which was the product by 300 and the sum 171,000 was 375 times 456. To complete the operation he multiplied 456 by 3, which produced 1368, and having added this number to 171,000, he found the result to be 172,368."

From this it appears that he was so little acquainted with the common rules as to multiply by 5 and then by 20, to find what the mere addition of two ciphers would have given him. In fact the whole operation seems awkwardly adapted to mental calculation; but with him it was probably nature's method, and it produced the sought for result.

We have not the full and satisfactory account of his constitution and habits that would be desirable; nor do we know any thing of his subsequent history. In order fully to appreciate such phenomena it is necessary to know more than merely the results produced by them. We have seen an account of a clerk in the war office, in France, who in six minutes extracted the square root of 20,511,841; and in a quarter of an hour, without any written memoranda, gave the product of 379,625,348 multiplied by itself. But we know nothing of him beyond this performance, and of course cannot class him with any other.

In 1845 a child named Prolongeau, aged about six years, was announced in the city of Paris, that resolved difficult arithmetical problems, and even elementary operations in algebra; and a committee was appointed by the Academy of Sciences to report the facts of the case, with his modes of operation, &c. His countenance is spoken of as expressive; but we have been unable to learn further particulars, or that the committee has reported.

George Bidder, a native of Devonshire in England, born in 1805, afforded another instance of extraordinary calculating powers when a mere child; and a number of gentlemen in Edinburg, undertook the charge of his education; with the design of cultivating his powers to the utmost extent. But though he excelled in Numbers, he proved nothing more than common in Geometry; and by no means realized the hopes of his friends. When only eleven years of age, he would solve difficult algebraical problems in a minute or two; but he failed in Geometry.

We might mention other instances noticed in books, but we have not such particulars as would enable us to give a satisfactory account of them; and after having mentioned two or three minor cases in our own country, we shall close with a somewhat detailed account of Truman Henry Safford, who differs from all the foregoing, and is perhaps the most remarkable character in this respect, known to be in existence.

An individual, named Peter M. Deshong, has been traversing the United States for several years past, who possesses an astonishing degree of quickness in performing the elementary operations of Arithmetic, and especially in adding numbers; but his knowledge seems limited to the mere elements of the subject. We saw him several years ago and again very recently and have no hesitation in saying that in adding together long columns of numbers, he very far exceeds in rapidity, any other person that we ever saw attempt the operation. His eye catches the numbers with the rapidity of thought, and he gives the result almost at a glance. In multiplying, he uses but a single line, however large the multiplier may be, and in dividing, he uses a mode very similar to short division, the remainders only being set down. He manifests unwillingness to engage in calculations involving intricacy, and we doubt his ability to reason to any considerable extent on the subject.

His practice is to travel from one important point to another, and exhibit his powers of calculation; at the same time offering for five or ten dollars to teach others to perform with equal rapidity. This he asserts he can do in half an hour; and to aid in the imposition he carries with him charts professing to give his modes of operation. Within a few years he has entirely changed his charts, and they are now well adapted to his purpose. Having carefully examined both his old and his revised charts, we have no hesitation in saying that he who expects to derive any thing valuable from them in regard to adding numbers, or from the instruction of their author, will find himself mistaken. His mode of multiplying is ingenious, and might be profitably employed by many; and to some extent the same is true of division; but his power of rapidly adding and otherwise combining simple numbers, is nature's gift as much as Zerah Colburn's was, and cannot be bought for money, nor acquired by any ordinary amount of practice. In addition to the peculiarities of nature, Mr. Deshong's whole time is devoted to these operations, and he has evidently improved by practice. Like all others whose minds are especially adapted to numbers, his memory on the subject is peculiarly retentive and prompt; and thus he is greatly aided in producing results. His bold and positive assertions are calculated to deceive many; but we understand thoroughly his professed modes of operation, and we have seen no one who has profited by his instruction in the addition of numbers. We say his professed mode, for we do not for one moment believe that he adds in the manner indicated by his charts. Addition may be thus performed, but the labor would be greater than in the ordinary way, and could not be performed so rapidly, unless when numbers are set down with special reference to that mode of addition. We have no wish to speak uncourteously of Mr. Deshong, but feel it to be our duty to warn the unwary, without wishing to prevent any one who desires to seek his instruction from doing so. For his mode of multiplying, see page 280.

We have received a detailed account of the peculiar powers of John Winn, formerly of Clark County, Ohio, and as the case differs from such as we have been considering, we shall give pretty free extracts from the letter before us.

"In person he was large, and in the latter part of his life corpulent. The features of his face were prominent, and indicated decision and determination. Whatever he undertook was pursued with ardor; and this remark applies as well to his religious and political opinions, as to his business transactions. He was decided in his friendships and his antipathies. His early education was limited, but as far as it went, was accurate and thorough. He was a good practical surveyor; his written compositions were free from errors in orthography or syntax, and his hand writing unusually neat, compact and uniform. Papers drawn by him were always executed in a business-like manner.

The most remarkable feature of his mind, however, was his facility in calculation. He was for some years engaged in buying and driving cattle and swine to market; and he prided himself on the rapidity and accuracy with which he could ascertain the numbers contained in droves, especially of swine. An opening would be made in a field containing a large drove, and he would sit on horseback, near the gap, and count as the animals were driven through, expressing himself audibly in something like this manner: "Twenty-five—sixty—eighty—rush them on boys!-hundred and twenty," and so on. Notwithstanding the rapidity with which he counted, he rarely ever macde a mistake; and his estimate was considered conclusive.

In adding multiplying and dividing numbers, he possessed uncommon facility. Instead of summing up units, tens, hundreds, &c. separately, as is usual in addition, he would run the whole up together with as little apparent trouble as a common operator would feel in adding up a single column. In multiplying or dividing by numbers of two or three places he took the whole together as we do numbers under 12. He was fluent in conversation, possessed a retentive memory; and with early discipline would have been capable of superior attainments.

We have met with a description of Georgr Blesins, son of John Blesins of Nashville, that would seem to rank him amongst the most remarkable prodigies of the present or the past; but we have been unable by writing, to learn any thing further respecting him.

Georgr Blesins is described as being about seven years of age, (in 1847) of common statue, in good health, and very interesting in his appearance and manners. His head is unusually large, his countenance one of those speaking ones that tell the fire within; while his whole demeanor is dignified and commanding. Our informant states that on asking him the product of 25 by 25, he answered instantly 625; and on being asked how he knew, he said "20 by 20 is 400; 5 by 20 is 100, and this doubled is 200; 5 by 5 is 25; and then 400 and 200 and 25 make 625."

"He was then asked how many inches there were around the globe. He replied that there is a certain number of inches in a mile, and this number multiplied by 25,000 will give the circumference in inches. While his thoughts were engaged in the calculation, there was considerable merriment among the company, which did not divert his attention the least. Some person spoke to him, to see what effect it would produce upon him. He replied, "Be patient a moment and then I will answer." Nothing could change the current of his thoughts when once put in motion. He in three minutes gave the exact distance round the globe, in inches; and this entirely by a mental process, for he knew nothing of figures."

Other instances of his calculations might be given, but we have not room. His powers of mind seem adapted to reasoning generally, and hence he belongs rather to the Safford than the Colburn school.

We shall close the notices of these cases with an account of Truman H. Safford, whom we shall notice somewhat fully.

Truman Henry Safford, Jr., is the son of Truman H. Safford, Esq., of Royalton, Vermont, where the son was born on the 6th of January, 1836. His frame is slight and his health has always been delicate, though he is represented as now acquiring greater strength. His hair and eyes are dark, and the latter shine with peculiar brilliancy; while his native modesty and kindness of manner, render him peculiarly interesting. His moral and reasoning faculties are astonishingly developed; but we might well say of his body, as a steamboat captain is reported to have exclaimed of John Quincy Adams, while contemplating that wonderful man as he stood, in venerable age, the centre of an admiring group, "O that we could take the engine out of the old Adams, and put it into a new hull!" But he that formed the brilliant machinery of young Safford's frail bark, can give it strength for his purpose in the hour of need.

At twenty months of age he had learned his letters, and already could be seen the workings of faculties that were soon to astonish every beholder. At three years he was familiar with many things seldom noticed by those of twice his age; and already, though but a prattling child whose tongue had but imperfectly learned the legerdemain (excuse the solecism) necessary to shape the words he used, his mind was breaking its fetters and struggling to understand the objects around him. He was sent to school, but the rules of study and of recitation were irksome to him, and he preferred to be at home where he could revel in study without control. In arithmetic he could not confine himself to the dull routine of the common rules and modes of operation. He saw the whole at a glance, and went through with a hop, skip and a jump, where others spent their days and weeks in slowly feeling their way. Instead of the neatly arranged rows of figures and the long columns that gradually step by step brings the result to the light of common minds, he would throw upon his slate a mass of half expressed numbers, in heterogeneous confusion, while his mind leaped beyond, and the conclusion was reached; but by giant strides that his teacher could not follow; and it was very soon concluded to leave him to his own course. His studies embraced every thing and any thing that came in his way—Geography, Chemistry, Grammar, and whatever afforded food for thought; and all were pursued with success.

The subjoined account of this wonderful youth was written in January, 1846, bythe Rev. Henry W. Adams, agent of the American Bible Society, and contains as full an account as may be necessary for our purpose. We may add that since Mr. Adams' article was written, ample provision has been made for the boy's education at Cambridge University, by the noble generosity of some public spirited friends of science; and he is now receiving every attention that can guard his health carefully, while he is enjoying the benefit of the greatest facilities that books, apparatus and living instructors can furnish. In order that his parents may watch over him, arrangements have been made to justify Mr. Safford in removing with his family to Cambridge and to support them for five years, during which time HENRY is to remain in that institution free of charge. Every precaution is used to protect his health, and for this purpose strangers are not allowed to visit him unless by express permission; while a board of physicians constantly guard against excess of study; and all tests of his powers, for the gratification of visiters are forbidden. From a friend who has the best opportunity of knowing, we learn that his general health and strength are improving under the judicious course pursued, and that he is rapidly advancing in his studies. Ile entered the University in September, 1846, and the guardians of his education furnish him with books and instruction, for five years at least; so that whatever may be the result, the friends of science will have the consolation of knowing that every effort has been made to foster the talent that now promises so much for the cause of human knowledge. We ought to remark that before the Cambridge arrangement was made, the youth calculated almanacs for 1846 and 1847, both of which were published, as well.as an edition for 1847, adapted to the latitude of Cincinnati, and published in that city. He was but little over nine years of age when the almanac for 1846 was calculated, and only ten when those for 1847 were calculated. This was certainly an effort of childhood that has no parallel.

We will now give Mr. Adams' account of his interview with the boy in January, 1846.

"Being a few days in the vicinity of Royalton, Vermont, on business connected with my Bible agency, I was induced, by the reports I had often seen in the public prints, of a remarkable boy of that town, to pay him a visit. The name of this precocious youth is Truman Henry Safford, Jr. At the age of twenty months he learned his letters. Before three years old, he would reckon time upon a clock almost intuitively. He also learned to enumerate according to the Roman method from Webster's spelling book. He commenced going to school when three years old, but this he did not like. Since then he has been but very little, and now goes none at all. His mode of study was perfectly unique. He did not pursue the common circuitous route to the results of study. Probably no college in the United States could instruct him much, if any. When he first began to go to school, his teachers could not comprehend his ways, nor instruct his infant mind. Every branch of study he could master alone, with rapidity and ease. He commenced Adams' New Arithmetic on Tuesday morning, and finished it completely on Friday night! And when he finishes a book it is done perfectly. He would not fully set down his sums, but cover his slate with a shower of figures, and at once bring out the answer. The teacher would look on in astonishment, unable to keep up with him, or to comprehend his operations, carried on in his mind with the rapidity of lightning, and then dashed upon the slate, no matter which end first. His thirst for all kinds of knowledge is very great. The whole circle of the sciences is as familiar to him as a household word. His father obtained for him Gregory's Dictionary of the Arts and Sciences, in three large volumes. This work, you know, is a vast encyclopedia of knowledge, treating briefly upon all branches of human knowledge. This was just the work he wanted; for an outline of any thing is enough—he can make the rest. It was this book that first gave him a taste for the higher mathematics. Here he found the definition of a logarithm, and from this alone, went on and made almost an entire table of them before ever seeing one. One day he went to his father and told him he wanted to calculate the eclipses and make an almanac! He said he wanted some books and instruments. His father tried to put him off; but the boy followed him into the fields and whithersoever he went, begging for books and instruments, with a most affecting importunity. Finally, his father promised to accompany him to Dartmouth College, and obtain for him, if possible, what he wanted. At this the boy was quite overjoyed; so much so, that when they hove in sight of the college, he cried out in raptures, "O, there is the college! there are the books! there are the instruments!" But they did not find all they wanted. At Norwich, however, they made up their complement. On coming home, the boy took Gummere's Astronomy, opened it in the middle, rolling it to and fro, and dashing through its dry and tedious formulas, went out at both ends. By the way, this is his usual mode of study. He does not begin any book at the beginning, but always in the middle, and then goes with a rush both ways. I asked him if, when he opened Gummere's Astronomy in the middle, he could comprehend those complicated formulas which depended on previous demonstrations. He replied, he could generally, but sometimes he "looked back a little." On arriving home, he projected several eclipses, and also calculated them through all their tedious operations by figures. This, as all mathematicians know, involves a knowledge of the labyrinths of mathematics, and also of formulas and processes most complicated and difficult. He has recently made an almanac for A. D. 1846. Two editions—the first of seven thousand copies and the second of seventeen thousand—have already been published and nearly all sold. In thle almanac are the calculations of two eclipses of the sun, wrought out wholly by its infant author, besides other valuable tables; especially one showing the amount of duties on wool, under all the tariffs since the formation of the government up to the act of 1842. This table the boy calculated alone. And that he calculated, without aid, the two eclipses of the sun, is attested by the published certificates of judges, lawyers, doctors, and clergymen.

Not satisfied with the old, circuitous process of demonstration, and impatient of delay, young Safford is constantly evolving new rules for abridging his work. He has found a new rule by which to calculate eclipses, hitherto unknown, so far as I know, to any mathematician. He told me it would shorten the work nearly one-third. When finding this rule, for two or three days he seemed to be in a sort of trance. One morning, very early, he came rushing down stairs, not stopping to dress himself, poured on to his slate a stream of figures, and soon cried out in the wildness of his joy, "O! father, I have got it! I have got it! it comes! it comes!" I questioned him respecting this rule. He commenced the explanation. His eyes rolled spasmodically in their sockets, and he explained his work with readiness. To hear him talk so rapidly, and yet so technically exact, and so far above the comprehension of all, save the most profound mathematician, put to flight all my doubts, and filled me with utter astonishment. He said he did not know as his new rule would work in all cases, but as yet it had. He also remarked that the nearer noon the eclipse came on, the easier it was to apply his rule. But young Safford's strength does not lie wholly in the mathematics. He has a sort of mental absorption. His infant mind drinks in knowledge as the sponge does water. Chemistry, botany, philosophy, geography and history, are his sport. It does not make much difference what question you ask him, he answers very readily. I spoke to him of some of the recent discoveries in chemistry. He understood them. I spoke to him of the solidification of carbonic acid gas by Professor Johnston, of the Wesleyan University. He said he understood it. Here his eyes flashed fire, and he began to explain the process.

When only four years old, he would surround himself upon the floor with Morse's, Woodbridge's, Olney's, Smith's, and Malte Brun's Geographies, tracing them through and comparing them, noting all their points of difference. His memory, too, is very strong. He has poured over Gregory's Dictionary of the Arts and Sciences so much, that I seriously doubt whether there can be a question, asked him, drawn from either of those immense volumes, that he will not answer instantly.

I saw the volumes and also noticed he had left his marks on almost every page. I asked to see his mathematical works. He sprung into his study and produced me Greenleaf's Arithmetic, Perkins' Algebra, Playfair's Euclid, Pike's Arithmetic, Davies' Algebra, Hutton's Mathematics, Flint's Surveying, the Cambridge Mathematics, Gummere's Astronomy, and several Nautical Almanacs. I asked him if he had mastered them all. He replied that he had. And an examination of him for the space of three hours convinced me he had; and not only so, but that he had far outstripped them. His knowledge is not intuitive. He is a pure and profound reasoner. In this he excels all other geniuses of whom I ever read. He can not only reckon figures in his mind with the rapidity of lightning, but he reasons, compares, reflects, and Nwades at pleasure through all the most abstruse sciences, and comprehends and reduces to his own clear and brief rules the highest mathematical knowledge. His mind is constantly active. No recreation or amusement can avail for any length of time to divert him from mental effort.

Being accompanied by Rev. C. N. Smith, of Randolph, Vt., who was acquainted with Mr. and Mrs. Safford, I had free access to the boy, and ample opportunity for a long and thorough examination. I went firmly expecting to be able to confound him, as I previously prepared myself with various problems for his solution. I did not suppose it possible for a boy of ten years only to be able to play, as with a top, with all the higher branches of mathematics. But in this I was disappointed. Here follow some of the questions I put to him, and his answers. I said, Can you tell me how many seconds old I was last March, the 12th day, when I was twenty-seven years old? He replied, instantly, "852,055,200." Then said I, The hour and minute hands of a clock are exactly together at 12 o'clock: when are they next together? Said he, as quick as thought, "1h. 5/11 m." And here I will remark, that I had only to read the sum to him once. He did not care to see it, but only to hear it announced once, no matter how long. Let this fact be remembered in connection with some of the long and blind sums I shall hereafter name, and see if it does not show his amazing power of perception and comprehension. Also, he would perform the sums mentally, and also on a slate, working by the briefest and strictest rules, and hurrying on to the answers with a rapidity outstripping all capacity to keep up with him. The next sum I gave him was this: A man and his wife usually drank out a cask of beer in 12 days; but when the man was from home, it lasted the woman 30 days: how many days would the man alone be drinking it? He whirled about, rolled up his eyes and replied, "20 days." Then said I, what are the values of x in the equation a²+b²—2bx+x²=(m²x²)/b²? He sprung to his slate, and dashed on a few figures, and replied in about a minute, x=n/(n²—m²) x (bn+√(a²m²+b²m²—a²n²)) He also gave the negative value of x.

Then said I, What number is that which, being divided by the product of its digits, the quotient is 3; and if 18 be added, the digits will be inverted? He flew out of his chair, whirled round, rolled up his wild, flashing eyes, and said, in about a minute, "24." Then said I, Two persons, A and B, departed from different places at the same time, and traveled towards each other. On meeting, it appeared that A had traveled 18 miles more than B; and that A could have gone B's journey in 153 days, but B would have been 28 days in performing A's journey. How far did each travel? He flew round tht room, round the chairs, writhing his little body as if in agony and in about a minute sprung up to me and said, "A traveled 72 miles and B 54 miles—did'nt they? Yes." Then said I, What two numbers are those whose sum, multiplied by the greater, is equal to 77; and whose difference, multiplied by the less, is equal to 12? He again shot out of his chair like an arrow, flew about the room, his eyes wildly rolling in their sockets, and in about a minute said, "4 and 7." Well, said I, the sum of two numbers is 8, and the sum of their cubes 153. What are the numbers? Said he instantly, "3 and 5." Now in regard to these sums, they are the hardest in Davies' Algebra. I have had classes of one hundred scholars who have not been able to perform several of them. But young Safford, at one reading, comprehended them at a flash, and returned, almost instantly, correct answers. He also gave me correct Algebraic formulas for doing them. Then I took him into Plane Trigonometry. Said I, In order to find the distance between two trees, A and B, which could not be directly measured, because of a pool which occupied the intermediate space, the distance of a third point, C, from each was measured, viz: C A=588 feet and C B=672 feet, and also the contained angle A C B=55° 40 min.; required the distance A B? He seized his slate, covered it with a group of figures, performed some of it mentally, and brought out the answer in about two minutes, saying, "592.967 feet." I then gave him this in the mensuration of surfaces: What is the area of a trapezoid whose parallel sides are 750 and 1225, and the altitude 1540? He walked rapidly across the floor, and whirled about to and fro, and replied," 1,520,750." Then, said I, if the diameter of the earth be 7921, what is the circumference? He said, instantly, "24,884.6136." To do this, he multiplied 7921 by 3.1416. This he did mentally quicker than I could write the answer. Then I gave him this: How many acres in a circular piece of ground whose circumference is 31.416 miles? He sprung on to his feet, flew round the room, and in a minute said, "50,265.6." Then, said I, required the number of acres of blue sky in an ellipse whose semi-axes are 35 and 25 miles? He began to walk the floor again, twisting his little body, and whirling his eyes spasmodically, and in about a minute said, "1,759,296 acres." How did you do it? said I. Said he, "Multiply the semi-axes together, and that product by 3.1416, and that product by 640." And did you perform the entire operation in your mind so soon? "Yes, sir." Then I took him into the mensuration of solids. Said I, what is the entire surface of a regular pyramid whose slant height is 17 feet, and the base a pentagon, of which each side is 33.5 feet? In about two minutes, after amplifying round the room, as is his custom, he replied, "3354.5558." How did you do it? said I. He answered, "Multiply 33.5 by 5, and that product by 8.5, and add this product to the product obtained by squaring 33.5, and multiplying the square by the tabular area taken from the table corresponding to a pentagon."

Now let it be remembered that this boy is only ten years old—that he did this sum for the first time in about two minutes, almost wholly in his head—and who can account for it? * * * * * * * * * * *

I asked him to give me the cube root of 3,723,875. He replied quicker than I could write it, and that mentally,' 155, is it not?" "Yes." Then said I, What is the cube root of 5,177,717? Said he,"173." Of 7,880,599? He instantly said, "199."

These roots he gave, calculated wholly in his mind, as quick as you could count one. I then asked his parents if I might give him a hard sum to perform mentally. They said they did not wish to tax his mind too much, nor often to its full capacity, but were quite willing to let me try him once. Then said I, Multiply, in your head, 365,365,365,365,365,365 by 365,365,365,365,365,365!! He flew round the room like a top, pulled his pantaloons over the top of his boots, bit his hand, rolled his eyes in their sockets, sometimes smiling and talking, and then seeming to be in agony, until, in not more than one minute, said he, "133,491,850,208,566,925,016,658,299,941,583,225!" The boy's father, Rev. C. N. Smith, and myself, had each a pencil and slate to take down the answer, and he gave it to us in periods of three figures each, as fast as it was possible for us to write them. And what was still more wonderful, he began to multiply at the left hand, and to bring out the answer from left to right, giving first, "133,491," &c. Here, confounded above measure, I gave up the examination. The boy looked pale and said he was tired. He said it was the largest sum he ever did! In conclusion, I am aware that this narrative is almost incredible. But let it be remembered that I went a skeptic, took a good witness with me, examined the boy carefully, and here pledge my sacred honor that all I have here stated is true. Rev. Mr. Smith, of Randolph, Vermont, is a witness to the correctness of this report. Further, if any are disposed to disbelieve Tmy statement, I beg them to make a tour to Royalton, Vermont, where they will find the boy and have an opportunity to examine him for themselves. I was informed that he had been offered $1000 a year to cast interest for a bank not far from his father's. Mr. Saffordl has received many urgent proposals to permit his wonderful son to be carried round the world for exhibition, but he will not consent. Gentlemen of wealth have offered pecuniary aid to furnish the boy with books; &c.; especially one of Cincinnati—the patron of the distinguished Powers.

HENRY W. ADAMS.

Concord, N. H. Jan. 1846.

In comparing the preceding cases, we find a great diversity of general intellect, and even diversity in the prominent feature, but in some respects a striking similarity exists. In all cases in which the power of calculation exists in an extraordinary degree, the ability to recollect numbers is found to exist also; and it is believed that it will prove true in every phase of mind. We find boys in every school that are dull in this subject, and others that are bright, and we find in regard to the former, that however the memory may be in other matters, it is difficult to cause them to remember rules involving arbitrary numbers, as .7854, 3.1416, &c.; while with the learner that delights in the subject, these numbers are remembered with ease. Something is no doubt to be placed to the difference in the ability to concentrate the attention, but this is not sufficient to account for all. There seems to be a natural difference, and this keeps pace with the power of calculation, from the dullest to the brightest specimen; and like the power itself it is improvable by exercise.

We have already alluded to the distinction that may be drawn between the cases that seem to possess an unsought and apparently unacquired power, and those that are the result of patient practice. The former shows itself most clearly in uneducated persons, who must of necessity contemplate numbers without the aid of figures, by modes of their own invention; while the latter pursue the modes common with others, and hence in former cases the results seem the more astonishing. But if an individual of ordinary powers were faithfully trained by either mode, it would be found that a degree of proficiency might be acquired, that no one would anticipate: and this would increase with the intensity of attention, which again would be proportioned to the proficiency; for we love best and attend most closely to that in which we can excel.

We have seen detailed accounts of a school kept by J. E. Lovell, Esq., at New Haven, Ct., in which mental operations in arithmetic are made a very prominent subject of study, and the power to which the pupils attain is almost incredible. The multiplying of fifteen or twenty places of figures by as many, is not unusual, the numbers being set down and the operation wrought mentally by cross multiplication. The surprise often expressed on witnessing the performances of arithmetical classes, when that science is made thle subject of especial study, would cease on a more intimate acquaintance with the powers of the human mind. Carry the calf daily and you may carry it when it becomes an ox. Mental performances, being more out of the usual routine of what is seen than written ones, excite most surprise; for he that is busied with other cares feels how impossible it is for him to turn away from the world and look in upon his own mind with the intense and unbroken gaze, indispensable to success; neither will they who now excite our surprise be able to do so when the cares and perplexities of life come upon them. Even Colburn, the wonder of the world, was unable to do his accustomed performances after other cares began to crowd upon him. In the case of Hagarman and some others to whom we have alluded, we find persons of maturity; but they were men with whom this was a constantly practised hobby. How far the object to be attained will justify the cultivation of this talent in youth, to the exclusion of others, is not the subject of our discussion. The man who appeared before a king of the olden time to show him with what certainty he could throw peas through the eye of a needle, had acquired astonishing dexterity; and the king duly appreciated his enterprise when he gave him a bag of peas for his pains. The thing can be done but is the acquisition worth the time and labor, and the sacrifice of other things involved?

If in other features the human mind presented no anomalies, her freaks in this particular would be more astonishing; but we find scarcely two minds constituted alike, or in which the several faculties are equally balanced. In our physical, mental and moral developments we find continual diversity; and while some manifest great deficiencies in one point, they exhibit perhaps equally astonishing prominence in others. Our time would fail to point out exemplifications, but any one can supply them in abundance.

In what respect are the gifts of Colburn, Bidder, Garry, &c., different from the endowments of the rest of mankind? Are they something distinct, or only extraordinary developments of what belongs to the human mind generally?

We have heard their peculiarities spoken of as instinctive; and have seen the term used in print in reference to them: but this cannot be a correct designation. It is doubtful whether instinct ever improves. We imagine that the first cell of the young bee is as perfect a hexagon as it can construct in old age; and we do not see that the young bird mistakes the materials of which its species usually build their nests, or constructs a different form. In every case of computing power to which we have alluded, and in every case in community the power is improved by cultivation, and lost by disuse. We do not for a moment believe that Colburn would have lost his ability, had he continued to cultivate it; and been freed from his pecuniary and other cares and perplexities.

The term intuitive, as applied to the case is less exceptionable; if we understand it to mean "Perceived by the mind immediately without the intervention of a train of reasoning or testimony. Perceived by bare inspection." But it is hard to say how far this is true; for the operations of the mind are often so rapid that the steps elude our observation, and we think we see at once, what indeed costs us a train of reasoning. The celebrated Dugald Stewart believed that all the conclusions of Colburn were reached by processes of reasoning, so rapid as to elude his own grasp, and to make no impression upon his memory; and hence he could give no account of them. After examining Colburn, Stewart seemed to attribute much of his peculiar power to Memory and Concentrated Attention; but these would produce rather small results when brought to bear on a mental blank. Yet there is no doubt but that a high degree of both was necessary to enable him to produce results so astonishing to the world.

It must be that a basis of axiomatic truths exists in all minds; and bare perception, intuition if you please, suffices to establish their character. But then the range and extent of these will depend on the ability of the mind to perceive; and the same mind after being cultivated, possesses greater ability to perceive and compare than when in a state of nature. It might be very difficult to decide where mere perception or intuition ceases, and reasoning commences, for they blend by shades so imperceptible, that there is no clearly defined line. They would seem to vary with different minds, and with the same mind under different circumstances. But whether the cases under consideration owe their peculiarity entirely to a perceptive power, beyond their fellow men, or alone to an ability to reason on the subject, with a celerity and accuracy peculiar to themselves, or to both combined they are equally interesting subjects of philosophical investigation.

In the case of Cap we find the power over numbers existing as almost the only representative of mind; while in Colburn we find it in connection with an ordinary development of the other faculties; and in Safford, we find it combined with an astonishing development of the whole mind. We regard these three as the most remarkable cases on record; while the others to which we have alluded, seem to fill the intermediate spaces and to show a gradual ascent from the lowest to the highest.

In the first we have an idiot, with no other faculty of the mind susceptible of education; and as the cares of the world are not likely ever to intrude, we may expect to see this power continue with him, and no doubt it might be increased under proper cultivation. In the second this ability perished or was choked by the growth of harassing cares and perplexities by which its unfortunate and highly sensitive possessor was weighed down, at an early age. He lost the indispensable power of withdrawing his attention from other things and turning it in upon itself. It would indeed have been strange had it been otherwise. Perhaps too the net work of forms thrown around his mental operations, in breaking him into the ordinary routine of study and school discipline, embarrassed the free operation of those modes peculiar to himself; and of course adapted to his own mind. Safford differs from all the others, possessing the natural power, in being able to perceive and announce truths as promptly as they could; and yet to follow his own mental operation and make it intelligible to others. This would favor the belief that however astonishing the aptitude might be, and however rapid the perceptions, they are still analogous to the every day operations of the mind, and differ only in degree from those of the dullest school boy. The talent was not found to exist in Safford, without previous indications of mind; for he reasoned, as well as perceived, from infancy. And it might be matter of doubt, whether the course he pursues in explaining to others, is always the original process of his own mind. In one respect these individuals seem to have resembled each other, and that was in the effect of their mental operations upon their bodies, producing violent contortions, and seeming to rack their whole systems. A similar expression of the eye is also spoken of and an acuteness in moral perception. How far these things may have been true of the others, we are not advised. In the case of Bidder and of Colburn, the ability seems to have been confined to Numbers, for neither one, though tried, excelled in Geometry; but Safford seems equally at home in either. From the account given of Winn, we think he might have acquired great proficiency as an engineer. His accuracy in estimating objects within the field of physical vision was not necessarily associated with his power of combining numbers; but taken together, they would have been invaluable to an engineer, or a field officer.

Garry seems to differ in some respects from all the others, but not materially so, and we must make allowances for difference in description, by different individuals. Though he thought he saw through no intervening medium, he admitted that in difficult problems "the ghosts moved more slowly and solemnly." To his mental vision, the sums of large numbers, and their various combinations, were as clearly present, as the sum of 3 and 4 would be to an ordinary mind.

But with all their powers, if it were sought to make a profound mathematician it would probably be better to take a subject of ordinary aptitude, with a sound mind in a sound body, and whose reasoning faculties are susceptible of healthful discipline.

It would be a pleasant task to pursue this subject much farther, and for this the material is ample; but if what has been hastily brought together shall lead inquiring minds to investigation, the object hoped for will be attained. Though such cases are rare, they are legitimate and important subjects of study. We have done no more on the present occasion than merely to throw out suggestions, which we hope others will improve. These anomalies are invaluable in the study of mind; like some species of mania, they exhibit the mental constitution in weak and strong lights, favorable to contemplation. In the well balanced mind, much of the internal working is concealed; but in the cases alluded to, the features of weakness and strength stand prominently out, and invite scrutiny.