[In ink:] By               
[In ink:] James Orchard Halliwell, Esq.         

      A very civil man, and an excellent scholar :  modest and respectful :  perfect in the
Latin tongue : an ingenious mechanist.—WHITELOCKE.




[In ink:] This tract was never published.

(image of page 1)

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LL that the Author has aimed at in the following pages has been to rescue from oblivion the actions and writings of a man, who ought long since to have held a distinguished situation in the scientific annals of Great Britain.
      Prior to the appearance of the biography of Sir Samuel Morland in Chalmers' Dictionary, no notice whatever had been taken of him : the reason of this neglect being probably owing in a great measure to the scarcity of his works. The above-mentioned article, though tolerably accurate with respect to his life, is very incorrect in other matters, as may be readily seen by comparing it with the following pages.
      A short tract on Arithmetic is inserted at the end, not that it has any reference to the subject of our memoir, but because it appeared to be a seasonable opportunity for rescuing a curious piece of antiquity from destruction.

Feb. 1838.

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SAMUEL MORLAND was the son of the Rev. Thomas Morland, of Sulhamstead-Bannister, near Reading in Berkshire, and born somewhere about the year 1625, as we learn from the preface to his "Urim of Conscience," published in 1695, where he says that he had then passed the seventieth year of his age.
      Little is known of his early life except that he was educated at Winchester school, whence he removed to the University of Cambridge, and, according to Cole,
1 to Magdalen College, where he became acquainted with Bishop Cumberland.2

   Athenæ Cant. vol. M.
    2    Payne's Life of Cumberland, p. 5.

He remained at Cambridge for nearly ten years, but never took any degree, and the probability is that his University career was very unprofitable.

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   Soon after his departure from college, he entered upon some political affairs, which brought him into public notice. In the month of November 1653, he was sent, with Whitelocke and a retinue of other gentlemen, on the famous embassy to the Queen of Sweden : they returned in July 1654, but, though successful, their rewards were anything but satisfactory. Morland, a few months afterwards, became assistant to Thurloe, the secretary of Oliver Cromwell. He also took a prominent part in the attempt to relieve the sufferings of the poor people of Piedmont, being appointed "commissioner extraordinary for the distribution of the collected monies" by the Protector, who also made him one of the clerks of the signet in March 1665.3 [This date is amended in pencil in the margin, with a question mark, indicating it could be 1655.]
      During the time he remained with Thurloe, he was an eye-witness of several plots concocted between that statesman and Cromwell.4 The most remarkable one to which he was privy, is that usually known as Sir Richard Willis's plot against the life of Charles the Second, and since it had a considerable influence on the future conduct of Sir Samuel, a full narration cannot be considered irrelevant in this place.
      In the beginning
5 of the year 1659, Thurloe, Cromwell, and Sir Richard Willis, formed a design of ruining the king at one blow, by sending

   MS. Harl. M.B. 7502, vii.
    4    Autobiography, MS. Lambeth, 931.
    5    Birch's Life of Thurloe.

p.7 / over messengers with plausible letters "to invite him to come over in a single ship, with only his two brothers and a few more, to a certain port in Sussex upon an appointed day, where they were promised to be received and supported by 500 foot at their first landing, and 2000 horse within one day after." This plot was discussed in Thurloe's office when Morland was at his desk apparently asleep: Welwood6

   Memoirs, ed. 1700, p. 11.

says that Cromwell, when he saw him, drew his sword, and was only dissuaded from despatching him on the spot by the earnest solicitation of Thurloe, who assured him that Morland had sat up two nights together, and was certainly fast asleep. Disgusted at this proceeding he immediately determined to divulge the plot to the king, which he did by means of one major Henshaw, who was then imprisoned in the tower. The king, being thus cautioned, answered that "he could not be ready so soon as the appointed day," which gave the three projectors some apprehension and suspicion of the discovery. Not being satisfied, however, with this answer, Willis was appointed to contrive other letters, urging his majesty "to use expedition, and not lose so fair an opportunity for his happy restoration." The king answered that he was not very well, or something that appeared so frivolous, that they justly concluded their whole project was discovered, and Willis was suspected p.8 / of having divulged it. Under these circumstances Willis sent for Morland, who went, not considering it safe to decline the meeting, but took two pocket - pistols with him. At the appointed place, he was met by another person, by whom he was conducted with the utmost caution into a dark deep cellar, where by the light of a candle he saw Sir Richard by himself with a bible before him. Sir Richard told him plainly that "he had sent for him on account of the discovery of a secret of the highest importance, which could not possibly be known to more than three persons besides himself." Then, recounting the particulars, he laid his hand upon the bible, and solemnly swore that he had not been the discoverer, and requested him to do the same. Morland, seizing one of his pistols, told him "he was ready to do so, if he would give him a reason why he should suspect him." All this he did with such a remarkable presence of mind that Willis was completely damped, and Morland escaped from further interrogation. In May 1660,7 he went to the king at Breda, who received him kindly, made him a knight, and soon afterwards a baronet.
8 produces a letter from Sir Samuel to Willis, dated March the 10th 1660, in which

   Lower's Journal, p. 12; Kennet's Register, p. 135.
    8    Hist. of England, p. 728; Harris's Life of Crom. vol. i. p. 229.

p.9 / he expressly denies the whole of the above. No credit, I conceive, must be given to it, since we have Morland's own testimony to the contrary in his autobiography : or, if he did write it at all, it was probably as a means of safety from the wrath of Willis.
      On the restoration of Charles he was made Master of Mechanics to his Majesty, who also presented him

   Evelyn's Numismata, p. 141.

with a medal as an "honorable badge of his signal loyalty." On one side of it was the king's head laureat, with this legend, "Carolo II. Regi institutori Aug.;" in the table of the reverse, "In adversis summo vitæ periculo in prosperis felici ingenio frequens adfuit." He was soon afterwards made a gentleman of his Majesty's privy chamber.
      During the remainder of his life he employed himself entirely with mechanical experiments, &c. In 1677 he took a lease of a house
10 at Vauxhall, for twenty-one years, from the heirs of Jane Vaux, the daughter of Guy Vaux, of gunpowder celebrity. This house was situated where Vauxhall Gardens now are.11

   On the top was a Punchinello holding a dial. (Aubrey's Surrey, vol. i. p. 12.)
    11    Manning's Surrey, vol. 3, p. 489.

Two years afterwards, he had a pension of 400 settled upon him, but some embarrassment in his affairs obliged him to sell it. He afterwards removed to a house at Hammersmith, near the water-side, p.10 / where he died December the 30th, 1695, and was buried in Hammersmith chapel on January the 6th of the following year.12

   Gent. Mag. 1818, part II. p. 12.

      The three last years of his life were spent very wretchedly. Poverty and loss of sight compelled him to rely almost solely on the charity of Archbishop Tenison. He returns him thanks for his kindness, in a letter dated March 5th, 1694, which was far greater, says Sir Samuel, "than such a poor wretch as I could ever hope for." Evelyn, in his diary, gives an interesting description of him when harassed with this accumulated load of misfortunes.
      " 25th Oct. 1695.—The Archbishop and myself went to Hammersmith to visit Sir Samuel Morland, who was entirely blind ; a very mortifying sight. He shewed us his invention of writing, which was very ingenious; also his wooden calendar, which instructed him all by feeling, and other pretty and useful inventions of mills, pumps, &c., and the pump he had erected, that serves water to his garden, and to passengers, with an inscription, and brings from a filthy part of the Thames near it, a most perfect and pure water. He had newly buried 200 worth of music-books, being, as he said, love songs and vanity. He plays himself psalms and religious hymns on the Theorbo."
      The inscription which Evelyn refers to was on a stone tablet fixed in the wall, and is still
p.11 / preserved ; the following is a copy of it: "Sir Samuel Morland's well, the use of which he freely gives to all persons: hoping that none who shall come after him, will adventure to incur God's displeasure, by denying a cup of cold water (provided at another's cost and not their own) to either neighbour, stranger, passenger, or poor thirsty beggar. July 8, 1695."
      Sir Samuel married three times : he was divorced from his last wife in 1688. One daughter of the name of Anne was buried at Kensington church, March 2, 1670, and another named Susanna was baptized at Norwood, February 28, 1666. The monuments of his two first wives are in Westminster Abbey.
      But Morland's principal claims to the notice of posterity are his writings and mechanical inventions : these I shall place as nearly as possible in chronological order, and I have endeavoured to be impartial, having given the various claims to the same inventions, which have been brought forward by others.
      From some correspondence between Morland and Dr. Pell, preserved in the British Museum,

   MS. Birch, 4279.

it appears that Sir Samuel, as early as 1666, had intended to publish a work on the quadrature of curvilinear spaces, and had actually proceeded to print part of it, when, by the advice of the latter, he was persuaded to lay it aside altogether. In the rough draft of a letter to p.12 / Morland, dated April the 7th, 1666, in giving his opinion on the portion of the work already printed, Pell says, "The love which I beare to truth and to the author of those papers does constraine me to desire that they may rest awhile unpublished." Morland not only yielded implicitly to his directions, but, in a letter written a short time afterwards, he furnishes argumentsagainst [as printed] some propositions in his own treatise : "I should desire," says he to Pell, "to bee altogether mute, and to submitt to your judgement in all things." Pell in another place14 informs us that Morland's "cyclometrical papers are two: one a copper-plate, the other a Latin discourse on it."15
      It was about this period that he invented his arithmetical machines, which he makes mention of in a letter dated May 13, 1666.

   MS. Birch, 4407.
    15    They became acquainted in Switzerland, but Morland tells us that he saw Pell for the first time at Colonel Montagu's chambers at Whitehall. See MS. Lansd. 751, folio 399.
    16    MS. Birch, 4279.

He did not, however, publish an account of them before 1673, when, "by the importunity of his very good friends," they were made public. The little work in which they are described is illustrated with twelve plates, in which the different parts of the machine are exhibited. Its operations are conducted by means of dial plates and small indices, moveable with a steel pin. By these means the four fundamental rules of p.13 / arithmetic are very readily worked, and, to use the author's own words, "without charging the memory, disturbing the mind, or exposing the operations to any uncertainty." His "perpetual almanac" is given at the end, which was reprinted by Playford in his "Vade Mecum."
      From Dr. Pell's collections in the British Museum,
17 it appears that Dr. Hooke had invented about the year 1670, an "engine for multiplying and dividing." The Marquis of Worcester18 also, in his Century of Inventions, seems to refer to a similar machine : he calls it "an instrument whereby persons ignorant in arithmetic, may perfectly observe numerations and subtractions of all sums and fractions."

   MS. Birch, 4422, folio 67. See also Hooke on the first part of Mach. Cœl. Hevelii, p. 45.
    18    Gent. Mag. 1748, p. 109.

      Most biographers assert that he invented the fire-engine, but he ought to be considered rather an improver than an inventor of that machine. As early as 1590, Cyprian Lucar, in his "Treatise named Lucarsolace," gave a description of a rude fire-engine which he designated by the name of a squirt : this engine consisted of a large hollow cone, moveable on a wooden frame, and open at the vertex, into which is inserted a long pipe for the passage of the water, which being continually thrown into the cone through a funnel near the vertex, is ejected by a piston at the other extremity, on a principle precisely similar to that p.14 / of the common squirt. Evelyn also mentions a fire-engine invented by Greatorix in 1656, which was ten years before he saw the "quench-fires" of Sir Samuel.
      We are, however, certainly indebted to Morland for the speaking-trumpet, an account of which instrument he published at London in 1671, under the title of "A description of the Tuba-stentorophonica, an instrument of excellent use, as well by sea as by land." In this rare tract, consisting of eight leaves, he gives an account of the various experiments that he made before his instrument attained a certain degree of perfection. The first trumpet that he constructed, although, says Sir Samuel, "the invention had been long before digested in my thoughts," was made in glass in the year 1670, being about 2 feet 8 inches in length, the diameter of the greater end 11 inches, and that of the other end 2½ inches : "with this," he says, "I was heard speaking at a considerable distance by several persons, and found that it did very considerably multiply the voice." After giving a description of some experiments with other trumpets, he enters into a philosophic disquisition on the nature of sound, and the best form of the speaking-trumpet, which he leaves doubtful, and concludes with "an account of the manifold uses" of his instrument, which are very excusably magnified : he appears also to have overrated the power of his trumpet, for, in his
p.15 / "Urim of Conscience," he says that he has no doubt but what it might be improved so as to carry the voice for the distance of ten miles. A French translation of Morland's tract was published at London in 1671, and, in an advertisement prefixed, it is stated that Morland's tubes were sold by Moses Pitt, a bookseller in St. Paul's church-yard, at the price of 2. 5s.
      The principal objects of Sir Samuel's study were water-engines, pumps, &c., which he carried to a high degree of perfection : his pumps brought water from Blackmore Park near Winkfield to the top of Windsor Castle. A bill to enable him "to enjoy the sole benefit of certain pumps and water-engines by him invented," was read the first and second times in the House of Commons, on the 12th and 13th of February 1674, but it did not pass : he obtained, however, a patent for them in the course of the following year. In 1697, two years after his death, a tract by him was published at the expence of his son : it is entitled "Hydrostatics; or, instructions concerning water-works," and contains an account of his various methods of raising water, besides tables of square and cube roots : from the close of Joseph Morland's preface, it appears that many of his father's works were still left unpublished. There is also a treatise by Sir Samuel, in the Harleian collection of manuscripts, which is entitled "Elevation des eaux, par toute sorte de machines, reduite a la mesure au poids, et
p.16 / a la balance : Presentée a sa majeste tres Chretienne, 1683 :" at page 35 commences a very short tract on the steam-engine, entitled "The principles of the new force of fire invented by Chev. Morland in 1682, and presented to his most Christian Majesty, 1683 ;" and these principles are explained as follows :— "Water being converted into vapour by the force of fire, these vapours shortly require a larger space (about 2000 times [this appears to be corrected in pencil in the margin to indicate '1000' or '7000' times] ) than the water before occupied, and rather than be constantly confined would split a cannon. But being duly regulated according to the rules of statics, and by science reduced to measure, weight, and balance, then they bear their load peaceably (like good horses) and thus become of great use to mankind, particularly for raising water, according to the following table, which shews the number of pounds that may be raised 1800 times per hour to a height of six inches by cylinders half filled with water, as well as the different diameters and depths of the said cylinders :" then follows his table of the effects of different sized cylinders.19

   Tredgold on the Steam-engine, p. 4.

This evidently indicates a knowledge of the subject, and we may, I think, fairly presume that he was probably the first who actually constructed a steam-engine, although his allusion to the force of steam being sufficient to burst a cannon appears to intimate that he was not a stranger to the volume which the Marquis of Worcester had p.17 / published some years previously. To his great credit also, let it not be forgotten that he has correctly stated the increase of volume which water occupies in a state of vapour, which must have been the result of experiment : his researches, however, seem to have had little influence on the progress of the practical application of steam.
      I shall now briefly notice his miscellaneous treatises. In 1658 he published his "History of the Evangelical Churches of Piedmont," which was drawn up at the request of Archbishop Usher, and contains a most exact geographical description of the place, and a faithful account of the doctrine, life, and persecutions of the ancient inhabitants ; it possesses a print of the author by Lombart after Lilly, with his arms underneath.

   Ar : on two bars sable, three leopards faces jessant, fleurs de lis. When he was made a Baronet they were altered to sable a leopard's head jessant a fleurs de lis, or.

According to Beughem, he wrote "Articles and rules for the better government of his majesty's forces by land during this present war;" but I have never seen it, nor do I know what war is referred to. His "Doctrine of interest both simple and compound," published in 1679, is a very praiseworthy little volume, and the tables are very accurately calculated, but his "New rule for the equation of payments" is erroneous : another tract by him, consisting of four leaves, and entitled "The Count of Pagan's p.18 / method of delineating all manner of fortifications (regular and irregular) from the exterior poligone reduced to English measure and converted into Hercotectonick lines," was published in 1672, in Venn's "Military and Maritime Discipline." The "Urim of Conscience," which has already been mentioned, was written during his blindness, and is a very singular piece of composition : it contains reflections on the fallen state and insignificance of man, and the uncertainty of life. By one of his letters, dated 28th of July, 1688, it appears also that he had an intention of publishing the first six books of Euclid for the use of public schools.
      Morland is said to have written a treatise on the barometer, which was answered by Lord North in another tract on the same subject : I have seen neither of them.
      He is also said to have invented the capstan to heave up anchors, but he must be considered rather an improver than the inventor of that machine : the same remark will apply to various other performances, which have at various times been attributed to him.

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To alle suche even nombrys the most have cifrys as to ten. twenty. thirtty. an hundred. an thousand and suche other. but ye schal vnderstonde that a cifre tokeneth nothinge but he maketh other the more significatyf that comith after hym. Also ye schal vnderstonde that in nombrys composyt and in alle other nombrys that ben of diverse figurys ye schal begynne in the ritht syde and so rekene backwarde and so he schal be wryte as thus—1000. the cifre in the ritht side was first wryte and yit he tokeneth nothinge no the secunde no the thridde but thei maken that figure of 1 the more signyficatyf that comith after hem by as moche as he born oute of his first place where he schuld yf he stode ther
p.20 / tokene but one. And there he stondith nowe in the ferye place he tokeneth a thousand as by this rewle. In the first place he tokeneth but hymself. In the secunde place he tokeneth ten times hymself. In the thridde place he tokeneth an hundred tymes himself. In the ferye he tokeneth a thousand tymes himself. In the fyftye place he tokeneth ten thousand tymes himself. In the sexte place he tokeneth an hundred thousand tymes hymself. In the seveth place he tokeneth ten hundred thousand tymes hymself, &c. And ye schal vnderstond that this worde nombre is partyd into thre partyes. Somme is callyd nombre of digitys for alle ben digitys that ben withinne ten as ix, viii, vii, vi, v, iv, iii, ii, i. Articules ben alle thei that mow be devyded into nombrys of ten as xx, xxx, xl, and suche other. Composittys be alle nombrys that ben componyd of a digyt and of an articule as fourtene fyftene thrittene and suche other. Fourtene is componyd of four that is a digyt and of ten that is an articule. Fyftene is componyd of fyve that is a digyt and of ten that is an articule and so of others   .  .  .  .  .  .   But as to this rewle. In the firste place he tokeneth but himself that is to say he tokeneth but that and no more. If that he stonde in the secunde place he tokeneth ten tymes himself as this figure 2 here 21. this is oon and twenty. This figure 2 stondith in the secunde place and therfor he tokeneth ten tymes himself and ten tymes 2 is p.21 / twenty and so forye of every figure and he stonde after another toward the lest syde he schal tokene ten tymes as moche more as he schuld token and he stode in that place ther that the figure afore him stondeth : lo an example as thus 9634. This figure of foure that hath this schape 4 tokeneth but himself for he stondeth in the first place. The figure of thre that hath this schape 3 tokeneth ten tyme himself for he stondeth in the secunde place and that is thritti. The figure of sexe that hath this schape 6 tokeneth ten tyme more than he schuld and he stode in the place yer the figure of thre stondeth for ther he schuld tokene but sexty. And now he tokeneth ten tymes that is sexe hundrid. The figure of nyne that hath this schape 9 tokeneth ten tymes more than he schulde and he stode in the place ther the figure of 6 stondeth inne for thanne he schuld tokene but nyne hundryd. And in the place that he stondeth inne nowe he tokeneth nine thousand. Alle the hole nombre of these foure figurys. Nine thousand sexe hundrid and foure and thritti.

      The foregoing tract is written on a single leaf of vellum, which was found loose in an ancient manuscript on Astronomy in my possession, and is curious inasmuch as it is the most ancient piece on arithmetic in the English language
p.22 / that has hitherto fallen under my observation : in the British Museum there are none written in English of a date prior to the sixteenth century.
      The division of numbers into digits, articules, and composites, belong probably to a very ancient period : I find it in a treatise on arithmetic mentioned by Wallis, and which is now in the Bodleian Library at Oxford : the first chapter of it is given here on account of its similarity with the above.

“   Hæc algorismus ars præsens dicitur : in quâ
Talibus Indorum fruimur bis quinque figuris.
      0 . 9 . 8 . 7 . 6 . 5 . 4 . 3 . 2 . 1.
Primaque significat unum, duo vero secunda ;
Tertia significat tria ; sic procede sinistra
Donec ad extremam venis, quæ cifra vocatur.
Quodcunque illarum si primo limite ponas
Simpliciter se significat ; si vero secundo
Se decies : sursum procedas multiplicando :
Nil cifra significat : dat significare sequenti.
Post hæc dicta, scias breviter quæ tres numerorum
Distinctæ species sunt.   Nam quidam digiti sunt ;
Articuli quidam ; quidam quoque compositi sunt.
Sunt digiti numeri qui semper infra decem sunt ;
Articuli decupli digitorum ; compositi sunt
Illi qui constant ex articulis digitisque.
Ergo proposito numero tibi scribere ; primo
Respicias quis sit numerus : quia si digitus sit
Uno figura satis ei ; sed si compositus sit,
Primo scribe loco digitum post articulumque
Si sit articulus, in primo limite cifram,
Articulum vero in limite scribe sequenti.
Septem sunt partes non plures istius artis.
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    Addere, subtrahere, duplareque dimidiare ;
Sextaque dividere est, sed quinta est multiplicare ;
Radicem extrahere pars septima dicitur esse.
Subtrahis aut addis a dextra vel mediabis.
A leva dupla, divide, multiplicaque ;
Extrahe radicem semper sub parte sinistra.”

      This tract has been ascribed to Sacro-Bosco, but on what ground I know not: a copious commentary on it in the British Museum was written in 1451, so that it must have been composed some considerable time before that period. It concludes thus : "Et sic finit tractatus de arte algorismi ad laudem et gloriam ipsius qui vivit et regnat in seculum seculorum. Amen. Scriptus per manum A. W. anno Domini millesimo CCCC. quingesimo primo."
      From the numerous MSS. of it which remain, we might be led to infer that it was of great antiquity, and a copy in Gale's collection ascribes it to Dionysius Exiguus : it has also been considered of English origin, because many transcripts possess notes on the margin in that language, but that is not at all conclusive : the number of foreign MSS. transported into England at that period was very great ; for instance, we are told by Daniel Morley, in the dedication of his treatise "De inferiori et superiori parte mundi," (MS. Arund. 377, folio 88.) to the bishop of Norwich, that he returned from Spain to England "cum preciosa multitudine librorum."
      The two commencing lines of the foregoing
p.24 / poem have given rise to much controversy among authors who have written on the introduction of the Arabic numerals into Europe : Professor Peacock considers it to belong to the fifteenth century, but I find it quoted in Sacro-Bosco's treatise on the same subject, which was composed about 1250, and therefore it must have been written before then. Moreover, in the library of Corpus Christi College, there is an ancient MS. of this poem in the French language.21
      A curious MS. on arithmetic in the library of Trinity College

   There are three MS. copies of it in the British Museum. Bib. Reg. 8 C. iv. et 12 E.I. i. and MS. Cott. Vitell. A. I. iv. which last contains only the first chapter as given above.
    22    MS. Gale. O. 2. 45. folio 37.

refers these numerals to a Chaldean origin : it is entitled "compendiosa subtilitas multiplicationis," and commences thus : "Cirta Algorismum primo docendum procedere : secundo addere : tertio subtrahere : quarto duplare : quinto dimidiare : sexto multiplicare : septimo et ultimo dividere. Processio est quod quelibet figura quo loco posita habeat significare. Sunt igitur novem figuræ que secundum Chaldeos sinistro sunt scribendæ. Quæ tales sunt . 9 . 8 . 7 . 6 . 5 . 4 . 3 . 2 . 1 .   Est autem decima figura que quamvis nichul significat per se tam secundum locum optinens aliis confert significare quæ talis est 0."
      The word Algorism is of Arabic origin, al-
p.25 / though stated by many early authors to have derived its name from Algus, its supposed in ventor.23

   John of Norfolk, in his "Summula progressionis," ascribes the invention of arithmetic to Algus : the author was of All Souls' College, at Oxford, as we learn from the close of his little treatise :—"Et sic perfectus iste est tractatus brevissimus in collegio animarum Oxoniæ Anno domini millessimo quadringentesimo quadragesimo quinto." MS. Harl. 3742.

It need scarcely be mentioned that in the middle ages all scientific knowledge was received from the Arabs, and I am acquainted with only one instance in which a mathematical author of that period has stated his means of information to be derived from a Greek source. The work to which I refer is a very ancient one on Astronomy, the commencement of which is as follows :—"Here begynnis ye wyse book of phylosophye and of astronomye conteyned and mad of XX wysest phylosoforys and astronomerys yat ever were syn ye world was begunne. That is for to saye in ye lond of grece For in yat lond an engelychman ful wys and wel vnderstandyng of phylosophye and astronomye stodyeth and compyled yis book out of grek into engelysch graciously."24

      Numerical signs, or rather contractions, bearing a slight similarity to the Arabic numerals, are found in a very singular MS.25 on Arithmetic, in the British Museum, which is said to belong p.26 / to the twelfth century.26

   MS. Sloane B. M. 2453. 965. Also Bib. Reg. 17A. XXXII. A copy more ancient than any of these is in the author's possession.
    25    MS. Arund. 343. i.
    26    Hallam's introduction to the literature of Europe during the Middle Ages, vol. i. p. 150.

They consist of nine forms, and have names written over each of them, "not Greek, or Latin, or Arabic or in any other known language," but having a little resemblance to the last. The principal deficiency is the absence of the cypher, but the writer was evidently acquainted with the decuple value which a digit receives by its situation on the left of another. I perfectly agree with Hallam, or rather the gentleman from whom he acknowledges to derive his information, in considering that in these contractions may be traced the elements of our own numerical notation.

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A P P E N D I X.

N the expectation of finding some more particulars relative to Morland's Cyclometrical treatise, all Pell's manuscripts in Dr. Birch's collection in the British Museum have been carefully examined, and since they contain a great quantity of papers interesting to the mathematician, but which, from the very brief description in the catalogue, cannot be generally known to be there, a short abstract of the contents of the more important cannot but be acceptable.
      Pell's MSS. in Ayscough's catalogue are numbered as follows: 4279, 4280, 4365, 4394, 4395, and from 4405 to 4431.
      No. 4279 contains original letters to Dr. Pell from John Davis, Morland, and others, with the rough draft of Pell's answers ; as also a letter from Walter Warner to Mr. Payne, dated Oct. 17th 1634. The next volume is filled principally with rough drafts of letters in answer to Cavendish and members of his own family. No. 4365 contains papers relating chiefly to Piedmont. No. 4394 consists of various loose papers, among which is one entitled "An inventorie of the papers of Mr. Warner," which John Collins, in a MS. note, states that he received from Dr. Thorndyke, December 14th, 1667. The only article deserving of notice in the following volume (4395) is a long letter from Henry Briggs to Pell, in which he mentions his "suddaine longe journey p.28 / into the Northe," and answers some enquiries concerning logarithms which Pell had made in a former letter.
      In No. 4405 is a curious tract on Logarithms by Pell, entitled "Imitatio Nepeiria :" at p. 132 is a letter dated from Cambridge, December 10th, 1672, which, though it contains no signature, must have been written by Newton, and is curious inasmuch as it mentions his "newe method" of fluxions. No. 4408, though stated to be written by Pell, is wholly in the hand-writing of Nathaniel Torporley, and contains notes on the variation of the compass, triangular numbers, optics, &c.; it appears

   Torporley has added the following note :—" It will do well in this forme. And I leave it to Mr. Warner's discretion, whether he think it fit to give this mention or not, because he seemed to doubt of it." He afterwards wrote a tract against Harriot's Algebra, the MS. of which is in Sion College, and a volume of mathematical papers headed in Reading's catalogue of that library as " Algebraicæ, tabulæ sinuum, &c." is in his hand-writing : he gave his library to the college in 1633.

also that the address "ad studiosos mathematicos," inserted at the end of Harriot's Algebra, was written by him: he quarrelled with Warner soon afterwards, and that probably occasioned his "Corrector analyticus artis posthumæ Thomæ Harrioti" in Sion College library, in which he calls Harriot "homo evanidus," meaning probably that his fame as a mathematician would not last for any length of time. The next volume is also by Torporley and contains solutions of algebraical and geometrical problems. No. 4410 contains Pell's original MS. of his letter to Theodore Haak concerning Easter, and from No. 4413, folio 24, it appears that the problem generally ascribed to Colonel Titus was proposed to Pell in 1649 by William Brereton, who very probably had it from Harriot. The remainder consist principally of imperfect mathematical papers, and, although they contain a few curious articles, yet they do not appear to be of sufficient importance to merit any further notice in this place : to the mathematical p.29 / antiquary I would recommend a careful examination of all of them, under the conviction that there is enough of what is really curious and important to repay him well for his labour.
      It may not be amiss here to enquire what became of Warner's papers mentioned at page 27. Pell in a letter

   MS. Birch, 4280. This letter is dated August, 7th 1644 from Amsterdam.

to Cavendish thus makes mention of them :—"Again as to Mr. Warner's analogickes of which you desire to know whether they be printed. You remember that his papers were given to his kinsman, a merchant in London, who sent his partner to bury the old man : himselfe being hindred by a politicke gout, which made him keepe out of their sight that urged him to contribute to the Parliaments assistance, from which he was exceedingly averse. So he was looked upon as one that absented himselfe out of malignancy, and his partner managed the whole trade. Since my comming over, the English merchants here tell me that both he and his partner are broken and now they both keepe out of sight, not as malignants but as bankrupts. But this you may better enquire among our Hamburgh merchants. In the meane time I am not a little afraid that all Warner's papers, and no small share of my labours therein, are seazed upon and most unmathematically divided betweene the sequestrators and creditors, who (being not able to balance the account where there appear so many numbers, and much troubled at the sight of so many crosses and circles in the superstitious Algebra and that blacke art of Geometry) will no doubt determine once in their lives to become figure-casters and so vote them all to be throwen into the fire ; if some good body doe not reprieve them for pyebottoms, &c., for which you know analogical numbers are incomparably apt, if they be accurately calculated." In 1641, Pell tells us that he received Warner's analogical tables from the p.30 / author, and he adds that "the booke itselfe was of large paper, length 11, breadth 9 inches : 100 pages."3
      In the British Museum
4 there are three tracts by Walter Warner, transcribed by Hunt. Smithson : the first is on tangents, the second on the "commixture of metalls," and the other is entitled " Radii optici definitiones," written in 1634, and is the identical MS. which Pell says5 he received from Cavendish in 1640.

   MS. Birch, 4411.
    4    MS. Harl. 6755. and 6756.
    5    MS. Birch, 4417. folio 34.

      Dr. Hutton tells us that Birch procured Pell's mathematical papers for the Royal Society, and that there were several treatises by Warner amongst them. I have not yet had an opportunity of ascertaining whether any such papers are in the possession of that Society, but I think it extremely probable that those which Hutton refers to are the same which are now in the British Museum, and that they were prevented by some accident from reaching their intended destination. But then, where are Warner's MSS. which are stated to be with them ? the only one that can be conjectured with any probability to be written by him, is a tract in No. 4409, "De analogicis canonibus."
      Newton's letter to Pell, mentioned at page 28, affords an additional proof for the priority of his claim to the invention of fluxions. After giving the fluxional method of finding the subtangent to a curve, he adds :—" This, sir, is one particular, or rather a corollary, of a general method which extends itselfe without any troublesome calculation, not only to the drawing tangents to all curve lines, whether geometrick or mechanick, or however related to straight lines or to other curve lines, but also to the resolving other abstruse kinds of problemes about the crookednesse, areas, lengths, centres of gravity of curves, &c. Nor is it (as Hudden's method de maximis et minimis, and consequently Slusius, his new me- p.31 / thod of tangents as I presume,) limited to æquations which are free from surd quantities. This method I have interwoven with that other of working in equations by reducing them to infinite series. I remember I once occasionally told Dr. Barrow, when he was about to publish his Lectures, that I had such a method of drawing tangents, but some divertisement or other hindered me from describing it to him."


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