929 



TAYLOR, BROOK, 



TAYLOR, ISAAC. 



930 



cussion, law of density of the atmosphere, refraction of light. The first 

 enunciation of the celebrated theorem is as follows : 



PROP. VII. THEOR. III. 



Sint a; et z quantitates duoo variabiles, quarum z uniformiter augetur 



per data iucremcnta z, et sit nz = v, v z = X *v z = "*>, et sic porr6. 



Turn dico quod quo tempore z crescendo fit z + v, x item crescendo fiet 



v v *v v v t> u v 



Corollary I. expresses the corresponding theorem for decrements. 

 COROLL. II. 



Si pro Incrementis evanescentibus scribantur fluxiones ipsis propor- 

 tionates, factis jam omnibus u t>, X v, ,v, n v, &c. sequahbus quo tempore 

 z \iniformiter fluendo fit s-\-v fiet x, 



v 3 



+ &c. 



. V 



x + x + x 



Iz 1.2 a 2 1.2.3z 3 



4- x 



vel mutato signo ipsius v, quo tempore z decrcscendo fiet z v, x de- 

 crcscendo fiet 



. . v .. v" .'. #'* 



- X X + X X - + &C. 



Iz 1.2s 2 1.2.3z 3 



Taylor does not make much use of his own theorem in the ' Methodus 

 Incremeutorum,' but he shows his command over it in the paper above 

 cited on the roots of equations, in which he extends Newton's method 

 to other than algebraical equations. 



One would have supposed that such a theorem as that of Taylor, 

 the instant it was proposed, would have been hailed as tha best and 

 most useful of generalisations. Instead of this, it sunk, or rather 

 never rose, till Lagrange pointed out its power. This is perhaps an 

 assertion which some may doubt : we proceed to make it good. The 

 first criticism upon the whole work (without a word about the theorem) 

 was that of Leibnitz, in a letter to John Bernoulli (June 1716, vol. ii., 

 p. 380, of their correspondence), and it will show of what sort of view 

 the neglect of this theorem was the consequence. The translation is as 

 follows : " I have received what Taylor calls his ' Method of Incre- 

 ments.' It is an application of the differential and integral calculus to 

 numbers, or rather to general magnitude. Thus the English have placed 

 the horses, according to the proverb, behind the cart. I began the 

 differential calculus from series of numbers .... and so came natu- 

 rally from the general calculus to the special geometrical or infinitesimal 

 calculus. They proceed the other way, because they have not the true 

 method of investigation It is written obscurely enough." Ber- 

 noulli answers (August 1716, p. 389): "I have at length received 

 Taylor's book. What, in the name of God, does the man -mean by the 

 darkness in which he involves the clearest things ! No doubt to con- 

 ceal his habit of thieving : as far as 1 can make it out, I see nothing but 

 what he has stolen from me, through his thick cloud of obscurity." 

 The notion of Leibnitz prevailed for a long time, and is not quite 

 extinct in our own day, though rapidly expiring ; the Differential 

 Calculus was to be used only as the medium in which pure algebra 

 was to be applied to geometry and physics ; and even a generalisation 

 of existing theorems, expressed in the language of that Calculus, was a 

 positively erroneous mode of proceeding. 



In Britain, two really great disciples of Taylor, soon appeared, 

 STIRLING and MACLAURIN. The first (' Meth. Diff.,' p. 102) repeated 

 the theorem as given by Taylor himself, and adds that Herman had 

 also given it in the Appendix to his 'Phoronomia;' and as this last 

 work was published in 1716, were Stirling's assertion true, Herman 

 must probably be considered an independent inventor. But on 

 examining the appendix to the ' Phoronomia ' (p. 393), to which Stirling 

 refers, we find only the theorem in book v., lemma 3, of the Principia, 

 and John Bernoulli's series for integration. Maclaurin (' Fluxions,' 

 1742, p. 610) proved Taylor's theorem again in the way which has 

 since become common. But both Stirling and Maclaurin use only a 

 particular case of Taylor's theorem, expanding not $ (x + z), but 

 <f> (0 + z), or expanding <f>s in powers of z. Neither thought he was 

 doing more than proving Taylor's theorem, and both attribute the 

 result to Taylor. Nevertheless this particular case has been since 

 called Maclaurin's theorem, though, if not Taylor's, it is Stirling's. 

 Maclaurin's book was, no doubt, more read than either of the other 

 two ; it was the answer to Berkeley's metaphysical objections, and 

 contained great power and vast store of instances ; and this may have 

 been the reason why a theorem which was best used in, and best 

 known by, Maclaurin's book, should be called after his name. It is 

 well that it should be so, or rather, it would be well that the develop- 

 ment of <f> (0 + z) in powers of z should be called by the name of 

 Stirling; for in truth the development of < (a + b) in powers of b is 

 one theorem or another in its uses, and iu the consequences it sug- 

 gests, according as a or 6 is looked at as the principal letter. 



In the interval between Taylor's death and Lagrange's paper in the 

 Berlin Memoirs for 1772, in which he fir.rt proposed to make Taylor's 

 theorem the foundation of the Differential Calculus, the theorem 

 was hardly known, and even when known, not known as Taylor's. 

 We cannot find it in Hodgson's Fluxions (1736), in Maria Agnesi's 



EIOG. HIV. VOL. V, 



Institutions (1748), in Landen's Residual Analysis (1764), in Simpson's 

 Fluxions (1737), iu Emerson's Increments (1763), in Emerson's 

 Fluxion? (1743), in Stone's Mathematical Dictionary (1743), nor 

 in the first edition of Montucla's History (1758). We have 

 examined various other places in which it thould be, without find- 

 ing it anywhere, except in the great French Encyclopaedia (article 

 ' Series '), and there we certainly did find it, mentioned only inci- 

 dentally, and attributed by no less a person than Condorcet to 

 D'Alembert. The Abbe" Bossut, who wrote the preliminary essay, 

 knew nothing about the theorem at that time; though afterwards, 

 when he published his history of mathematics, he was better in- 

 formed. We found afterwards that Condorcet (Lacroiz, torn, iii., 

 p. 396) was in the habit of assigning this theorem to D'Alembert ; 

 not with any unfair intention, but in pure ignorance. The fact was 

 that D'Alembert (' llechercb.es sur difie'rens points,' &c., vol. i, p. 50, 

 according to Lacroix) gave for the first time the theorem accompanied 

 by a method of finding the remnant of Taylor's series after a certain 

 number of terms have been taken ; and Condorcet, who probably had 

 never seen the theorem elsewhere, thought it was D'Alembert's. In 

 fact, D'Alembert himself gave the theorem as if it were new, and 

 without mentioning the name of any one, which Lacroix says is ' assez 

 singulier,' an opinion in which we cannot agree. Unless D'Alembert 

 read English, we cannot imagine how he should have known Taylor's 

 theorem, nor even then, unless Taylor, Stirling, Maclauriu, or an old 

 volume of the ' Philosophical Transactions,' be supposed to have fallen 

 in his way. We have no doubt that D'Alembert was a new discoverer 

 of the theorem, and that Condorcet never saw it except in his writings. 

 Our wonder rather is where Lagrange could have found the name of 

 Taylor in connectic n with it. From Lagrange's time Taylor's theorein. 

 takes that place which, if it had always occupied, we should not have 

 had to write any history of it. 



*TAYLOR, HENRY, English poet, was born in the early part of 

 the present century. A great portion of his life has been spent as a 

 civil servant in the department of the Colonial Office ; in which office 

 he now holds one of the five senior clerkships. His first known 

 literary effort was ' Isaac Comneuus,' a play in five acts and in verse, 

 published in 1827. This was followed in 1834 by his more celebrated 

 poem 'Philip Van Artevelde, a Dramatic Romance, in two parts,' of 

 which there have been six or seven editions, and which has been 

 translated into German. In 1836 he published a prose work of a 

 different character, entitled ' The Statesman,' embodying, in the shape 

 of reflection, much of his experience of public and administrative life. 

 To this succeeded, iu 1842, 'Edwin the Fair, an Historical Drama,' 

 in five acts and in verse. In 1847, he published ' The Eve of the Con- 

 quest and other Poems ; ' and in the same year a prose work entitled 

 ' Notes from Life, in Six Essays,' of which there have been three 

 subsequent editions. In 1849 he gave to the world another work of a 

 similar character, entitled ' Notes from Books, in four Essays ; ' and in 

 1850 he published ' The Virgin Widow,' a play in five acts, and chiefly 

 in verse. There have also been collected editions of his poetical 

 writings. His various works have given him a high reputation with 

 the judicious as a man of thought and scholarship, and more par- 

 ticularly as a dramatic poet of great and peculiar ability one of the 

 few English poets of our time who have produced sterling poetic 

 works of a thoroughly English character in the dramatic form once so 

 dominant in our literature. 



* TAYLOR, ISAAC, the author of the 'Natural History of Enthu- 

 siasm ' and ' Ancient Christianity,' belongs to a family, several of the 

 members of which have honourably distinguished themselves, and of 

 whom we prefix a brief notice. 



ISAAC TAYLOR, senior, was a man of great decision of purpose and 

 of considerable mental power. Originally a line-engraver, he in 1786 

 removed from the metropolis to Laveuham, Suffolk, for the purpose 

 of pursuing his profession, and at the same time training his children 

 under his own eye in a quiet country town. Being a man of strong 

 religious feelings he was led to take an active part in the community 

 to which he belonged, and his occasional religious addresses being 

 found very acceptable, he was eventually, 1796, invited to become the 

 minister of an Independent congregation at Colchester, Essex. In 

 1810 he removed, on the invitation of a similar congregation, to 

 Ongar, where he remained till his death, December 11, 1S29. During 

 this period he not only laboured diligently in his ministerial calling, 

 and carefully educated his children, but likewise found time to write 

 numerous small books on educational subjects : ' Advice to the Teens,' 

 ' Scenes for Tarry-at-Honae Travellers,' 'Beginnings of Biography,' &c., 

 which had an extensive circulation. He also published a few sermons 

 and other religious works. His elder brother CHARLES TAYLOR (who 

 died November 1821), was the editor of Calmet's 'Dictionary of 

 the Bible,' and the author of 'Fragments' on subjects of Biblical 

 exposition. 



ANN TAYLOR, the wife of Isaac Taylor of Ongar, was likewise a 

 woman of superior ability and attainments. Having herself com- 

 pleted the education of her daughters, she somewhat late in life took 

 up her pen, and wrote several volumes very popular in their day 

 ' Maternal Solicitude,' and others, chieflv of an educational character. 

 She died in 1830. *ANN TAYLOR (Mrs/Gilbert), and JANE TAYLOR, 

 daughters of the above Isaac and Ann Taylor, also became widely 

 known as the authors of juvenile works of more than ordinary excel- 



30 



