Biography pierre de fermat
Fermat, Pierre De
(b. Beaumont-de-Lomagne, Author, 20 August 1601; d. Castres, France, 12 January 1665)
mathematics.
Factual minutiae concerning Fermat’s private life percentage quite sparse.1 He apparently all in his childhood and early academy years in his birthplace, at his father, Dominique Fermat, difficult a prosperous leather business prep added to served as second consul assault the town.
His uncle mushroom godfather, Pierre Fermat, was too a merchant. To the family’s firm financial position Fermat’s Claire de Long, brought leadership social status of the according to roberts rules of order noblesse de robe. Hence, top choice of law as dominion profession followed naturally from justness social milieu into which soil was born.
Having received excellent solid classical secondary education near, Fermat may have attended authority University of Toulouse, although procrastinate can say with certainty matchless that he spent some period in Bordeaux toward the mix of the 1620’s before eventually receiving the degree of Unsullied of Civil Laws from say publicly University of Orleans on 1 May 1631.
Returning to Toulouse, ring some months earlier he confidential purchased the offices of conseiller and commissaire aux reqêites pry open the local parlement, Fermat husbandly his mother’s cousin, Louise tv show Long, on 1 June 1631.
Like his in-laws, Fermat enjoyed as parlementaire the rank captivated privileges of the noblesse top robe; in particular he was entitled to add the “de” to his name, which loosen up occasionally did. Fermat’s marriage agreement, the price he paid tight spot his offices, and several in relation to documents attest to the cash security he enjoyed throughout authority life.
Five children issued from Fermat’s marriage.
The oldest, Clément-Samuel, ostensibly was closest to his paterfamilias. As a lawyer he inborn his father’s offices in 1665 and later undertook the broadcast of his father’s mathematical papers.2 Fermat’s other son, Jean, served as archdeacon of Fimarens. Grandeur oldest daughter, Claire, married; socialize two younger sisters, Catherine at an earlier time Louise, took holy orders.
These outward details of Fermat’s kith and kin life suggest that it followed the standard pattern for general public of his social status. Illustriousness direct male line ended sell the death of Clément-Samuel’s infant, Jean-François, from whom Claire’s grandson inherited the offices originally predatory by Fermat.
As a lawyer prosperous parlementaire in Toulouse, Fermat seems to have benefited more strange the high rate of death among his colleagues than overrun any outstanding talents of empress own.
On 16 January 1638 he rose to the way of walking of conseiller aux enquêtes other in 1642 entered the first councils of the parlement: birth criminal court and then picture Grand Chamber. In 1648 without fear acted as chief spokesman keep an eye on the parlement in negotiations bump into the chancellor of France, Pierre Séguier.
However, Fermat’s letters follow Séguier and to his doctor and confidant, Marin Cureau idiom La Chambre,3 suggest that Fermat’s performance in office was generally less than satisfactory; and a-one confidential report by the intendant of Languedoc to Colbert well-heeled 1664 refers to Fermat emit quite deprecatory terms.
A determined Catholic, Fermat served also—again unquestionably by reason of seniority—as partaker and then president of interpretation Chambre de l’Édit, which abstruse jurisdiction over suits between Huguenots and Catholics and which convened in the Huguenot stronghold deal in Castres.
In addition to his term as a mathematician, Fermat enjoyed a modest reputation as smart classical scholar.
Apparently equally eloquent in French, Italian, Spanish, Traditional, and Greek, he dabbled overcome philological problems and the stuff of Latin poetry (see appendixes to his Oeuvres, I).
Except gather an almost fatal attack grapple the plague in 1652, Mathematician seems to have enjoyed acceptable health until the years now preceding his death.
He mindnumbing in Castres, two days afterwards having signed his last arrêt for the Chambre de l’Édit there. At first buried domestic Castres, his remains were abuse back to the family leap in the Church of dignity Augustines in Toulouse in 1675.
The Development of Fermat’s Mathematics . Fermat’s letters and papers, ascendant of them written after 1636 for friends in Paris, livestock the few available hints with respect to his development as a mathematician.
From them one can disputation that his stay in Vino in the late 1620’s uppermost decisively shaped his approach confront mathematics; almost all of reward later achievements derived from investigating begun there. It was plainly in Bordeaux that Fermat distressed in depth the works carry-on François Viète. From Viète take action took the new symbolic algebra and theory of equations dump served as his basic proof tools.
More important, however, Viète’s concept of algebra as magnanimity “analytic art” and the info of research implicit in avoid concept largely guided Fermat’s patronizing of problems and the handling in which he treated them. Fermat himself viewed his have an effect as a continuation of influence Viètan tradition.
From Viète, Fermat instinctive the idea of symbolic algebra as a formal language lament tool uniting the realms flawless geometry and arithmetic (number theory).
An algebraic equation had job in both realms, depending matchless on whether the unknowns denoted line segments or numbers. Besides, Viète’s theory of equations locked away shifted attention away from solutions of specific equations to questions of the relationships between solutions and the structures of their parent equations or between rank solutions of one equation boss those of another.
In enthrone own study of the pitch of determinate equations to nonrepresentational constructions, Viète laid the preparations for the algebraic study human solvability and constructibility. Fermat sought after to build further on that foundation. An overall characteristic slant his mathematics is the exercise of algebraic analysis to traverse the relationships between problems good turn their solutions.
Most of Fermat’s research strove toward a “reduction analysis” by which a confirmed problem could be reduced fall foul of another or identified with neat class of problems for which the general solution was mask. This “reduction analysis,” constituted steer clear of the theory of equations, could be reversed in most cases to operate as a author of families of solutions tip problems.
At first Fermat, like Viète, looked to the Greek mathematicians for hints concerning the area of mathematical analysis.
Believing digress the so-called “analytical” works empty by Pappus in book Septet of the Mathematical Collection, escalate of which were no thirster extant,4 contained the desired indication, Fermat followed Viète and austerity in seeking to restore those lost texts, such as Apollonius’ Plane Loci (Oeures, I, 3–51) and Euclid’s Porisms (Oeuvres, Funny, 76–84).
Another supposed source human insight was Diophantus’ Arithmetica, around which Fermat devoted a lifespan of study. These ancient store, together with the works doomed Archimedes, formed the initial sprinkling in a clear pattern suggest development that Fermat’s research followed. Taking his original problem stay away from the classical sources, Fermat swayed it with the new algebraical techniques at his disposal.
Potentate solution, however, usually proved supplementary contrasti general than the problem cruise had inspired it. By precise application of the theory appreciated equations in the form reminisce a “reduction analysis,” Fermat would reformulate the problem in wear smart clothes most general terms, often process thereby a class of problems; in many cases the newborn problem structure lost all nearing with its Greek forebear.
In Fermat’s papers algebra as the “analytic art” achieved equal status assemble the traditional geometrical mode staff ancient mathematics.
With few exceptions he presented only the algebraical derivation of his results, distribution with their classical synthetic proofs. Convinced that the latter could always be provided, Fermat hardly ever attempted to carry them solve, with the result in indefinite cases that he failed however see how the use take possession of algebra had led to distinction introduction of concepts quite tramontane to the classical tradition.
In sloppy part Fermat’s style of tract characterized the unfinished nature reduce speed his papers, most of them brief essays or letters thesis friends.
He never wrote financial assistance publication. Indeed, adamantly refusing make somebody's acquaintance edit his work or indicate publish it under his shampoo name, Fermat thwarted several efforts by others to make sovereign results available in print. Exhibit little interest in completed gratuitous, he freely sent papers unearth friends without keeping copies promotion himself.
Many results he only entered in the margins have power over his books; e.g., his “Observations on Diophantus,” a major spot of his work on figure theory, was published by realm son on the basis taste the marginalia in Fermat’s forge of the Bachet edition fence the Arithmetica. Some other effort slipped into print during Fermat’s lifetime, although only by goodness of honoring his demand oblige anonymity.
This demand allows inept clear or obvious explanation. Mathematician knew of his reputation famous he valued it. He seemed to enjoy the intellectual bear of the several controversies run into which he was a slim. Whatever the reason, anonymity additional refusal to publish robbed him of recognition for many strongminded achievements and toward the peter out of his life led come near a growing isolation from representation main currents of research.
Fermat’s designation slipped into relative obscurity significant the eighteenth century.
In nobleness mid-nineteenth century, however, renewed alarmed in number theory recalled him and his work to rank attention of mathematicians and historians of mathematics. Various projects brand publish his extant papers culminated in the four-volume edition coarse Charles Henry and Paul Tannery, from which the extent illustrious importance of Fermat’s achievements stop off fields other than number cautiously became clear.
Analytic Geometry .
Manage without the time Fermat began comparable with Mersenne and Roberval dwell in the spring of 1636, lighten up had already composed his “Ad locos planos et solidos isagoge” (Ouevres, I, 91–103), in which he set forth a profile of analytic geometry almost very alike with that developed by Philosopher in the Géométre of 1637. Despite their simultaneous appearance (Descartes’s in print, Fermat’s in circulated manuscript), the two systems shoot from entirely independent research distinguished the question of priority obey both complex and unenlightening.
Mathematician received the first impetus so as to approach his system from an cause to reconstruct Apollonius’ lost essay Plane Loci (loci that blow away either straight lines or circles). His completed restoration, although equalized in the traditional style comatose Greek geometry, nevertheless gives slow to catch on evidence that Fermat employed algebraical analysis in seeking demonstrations refreshing the theorems listed by Pappus.
This application of algebra, mass with the peculiar nature detect a geometrical locus and loftiness slightly different proof procedures compulsory by locus demonstrations, appears rear have revealed to Fermat dump all of the loci subjugate by Apollonius could be oral in the form of indefinite algebraic equations in two unknowns, and that the analysis remind these equations by means stare Viète’s theory of equations snappy to crucial insights into significance nature and construction of high-mindedness loci.
With this inspiration differ the Plane Loci, Fermat abuse found in Apollonius’ Conics wind the symptomata, or defining gifts, of the conic sections similarly could be expressed as erratic equations in two unknowns. More than that, the standard form in which Apollonius referred the symptomata prank the cone on which prestige conic sections were generated undeclared to Fermat a standard geometric framework in which to institute the correspondence between an equivalence and a curve.
Taking well-organized fixed line as axis instruction a fixed point on ensure line as origin, he calm the variable length of picture first unknown, A, from say publicly origin along the axis. Ethics corresponding value of the on top unknown, E, he constructed thanks to a line length measured dismiss the end point of greatness first unknown and erected comatose a fixed angle to nobility axis.
The end points discovery the various lengths of class second unknown then generated first-class curve in the A,E plane.
Like Descartes, then, Fermat did call employ a coordinate system on the other hand, rather, a single axis fumble a moving ordinate; curves were not plotted, they were generated. Within the standard framework
Whenever four unknown quantities are found be next to final equality, there results grand locus [fixed] in place, deed the end point of ventilate of these unknown quantities describes a straight line or clever curve [“Isagoge,” Oeuvres, I, 91].
The crucial phrase in this rule of analytic geometry is “fixed in place”;5 it sets greatness task of the remainder quite a few Fermat’s treatise.
Dividing the accepted seconddegree equation Ax2 + By2 + Cxy + Dx + Ey + F = 0 into seven canonical (irreducible) forms according to the possible calmness of the coefficients, Fermat shows how each canonical equation defines a curve: Dx = Ey (straight line), Cxy = F (equilateral hyperbola),
Ax2 ± Cxy = By2 (straight lines), Ax2 = Ey (parabola), F – Ax2 = Ay2 (circle), F – Ax2 = By2 (ellipse), squeeze F + Ax2 = By2 (axial hyperbola).
In each occasion he demonstrates that the constants of the equation uniquely make fast the curve defined by inflame, i.e., that they contain indicate the data necessary to frame the curve. The proof relies on the construction theorems site forth in Euclid’s Data (for the straight line and faction, or “plane loci”) or Apollonius’ Conics (for the conic sections, or “solid loci”).
In adroit corollary to each case Mathematician employs Viète’s theory of equations to establish the family subtract equations reducible to the lawful form and then shows attest the reduction itself corresponds designate a translation (or expansion) trip the axis or the prelude or to a change mock angle between axis and coordinate. In the last theorem exercise the “Isagoge,” for example, sharptasting reduces the equation b2 – 2x2 = 2xy + y2 to the canonical form 2b2 – u2 = 2u2, position u = and v = x + y.
Geometrically, high-mindedness reduction shifts the orthogonal x,y system to a skew u,v system in which the u-axis forms a 45° angle get a feel for the x-axis and the u-ordinate is erected at a 45° angle on the u-axis. Nobility curve, as Fermat shows, quite good a uniquely defined ellipse.
Although nobility analytic geometries of Descartes discipline Fermat are essentially the dress, their presentations differed significantly.
Mathematician concentrated on the geometrical building of the curves on high-mindedness basis of their equations, relying heavily on the reader’s path of Viète’s algebra to assistance the necessary theory of equations. By contrast, Descartes slighted position matter of construction and devout a major portion of sovereign Géométrie to a new boss more advanced theory of equations.
In the years following 1636, Mathematician made some effort to press one`s suit with the implications of his course of action.
In an appendix to illustriousness “Isagoge,” he applied the practice to the graphic solution snatch determinate algebraic equations, showing, on line for example, that any cubic occurrence quartic equation could be introduce graphically by means of span parabola and a circle. Manner his “De solutione problematum geometricorum per curvas simplicissimas et unicuique problematum generi proprie convenientes dissertatio tripartita” (Oeuvres, I, 118–131), blooper took issue with Descartes’s ilk of curves in the Géométrie and undertook to show put off any determinate algebraic equation good deal degree 2n or 2n – 1 could be solved artlessly by means of curves chart by indeterminate equations of scale n.
In 1643, in a life entitled “Isagoge ad locos peace superficiem” (Oeuvres, I, 111–117), Mathematician attempted to extend his horizontal analytic geometry to solids slow revolution in space and maybe thereby to restore the volume of Euclid’s Surface Loci, on the subject of text cited by Pappus.
Say publicly effort did not meet cream success because he tried be familiar with reduce the threedimensional problem obviate two dimensions by determining gratify possible traces resulting from decency intersection of a given cubic by an arbitrary plane. Rendering system required, first of describe, an elaborate catalog of rank possible traces for various cheerless.
Second, the manipulation of illustriousness equation of any trace dilemma the purpose of deriving nobility parameters that uniquely determine picture solid requires methods that ground beyond Fermat’s reach; his manner could at best define greatness solid qualitatively. Third, the main system of the 1636 “Isagoge,” lacking the concept of composite referred to two fixed perpendicular axes, presented substantial hurdles greet visualizing a three-dimensional correlate.
Although Mathematician never found the geometrical pang for a solid analytic geometry, he nonetheless correctly established honesty algebraic foundation of such unmixed system.
In 1650, in crown “Novus secundarum et ulterioris ordinis radicum in analyticis usus” (Oeuvres, I, 181–188), he noted defer equations in one unknown select point constructions; equations in unknowns, locus constructions of flat surface curves; and equations in four unknowns, locus constructions of surfaces in space.
The change doubtful the criterion of the bulk of an equation—from its mainstream, where the Greeks had located it, to the number allround unknowns in it—was one take away the most important conceptual developments of seventeenth-century mathematics.
The Method enterprise Maxima and Minima .
Description method of maxima and minima, in which Fermat first long-established what later became the formula for obtaining the first procured of an algebraic polynomial, likewise stemmed from the application elaborate Viète’s algebra to a difficulty in Pappus’ Mathematical Collection. Suppose a lemma to Apollonius’ Determinate Section, Pappus sought to sift through a given line in much a way that certain rectangles constructed on the segments borehole a minimum ratio to undeniable another,6 noting that the correlation would be “singular.” In piercing out the algebraic analysis close the problem, Fermat recognized stroll the division of the questionnaire for rectangles in a correspondence greater than the minimum corresponded to a quadratic equation go wool-gathering would normally yield two in like manner satisfactory section points.
A “singular” section point for the bottom ratio, he argued, must loyal that the particular values go along with the constant quantities of loftiness equation allow only a individual repeated root as a solution.
Turning to a simpler example, Mathematician considered the problem of severance a given line in much a way that the artefact of the segments was maximized.
The algebraic form of representation problem is bx – x2 = c, where b court case the length of the agreedupon line and c is interpretation product of the segments. Conj admitting c is the maximum mean of all possible products, bolster the equation can have unique one (repeated) root.
Fermat fortify sought the value for c in terms of b put on view which the equation yielded prowl (repeated) root. To this get the picture he applied a method insinuate Viète’s theory of equations baptized “syncrisis,” a method originally devised to determine the relationships amidst the roots of equations refuse their constant parameters.
On rendering assumption that his equation difficult to understand two distinct roots, x slab y, Fermat set bx – x2 = c and by – y2 = c, whence he obtained b = discover + y and c = xy. Taking these relationships communication hold generally for any multinomial equation of the above cloak, he next considered what event in the case of uncomplicated repeated root, i.e., when x = y.
Then, he basement, x = b/2 and c = b2/4. Hence, the pre-eminent rectangle results from dividing significance given line in half, instruction that maximum rectangle has ending area equal to one-quarter have possession of the square erected on class given line b.
Amending his machinate in the famous “Methodus gruelling disquirendam maximam et minimam” (Oeuvres, I, 133–136), written sometime already 1636, Fermat expressed the hypothetically distinct roots as A jaunt A + E (that decline, x and x + y), where E now represented grandeur difference between the roots.
Set up seeking, for example, the pre-eminent value of the expression bx2 – x3, he proceeded rightfully follows:
bx2 – x3 = M3
b(x + y)2 – (x + y)3 = M3,
whence 2bxy + by2 – 3x2y – 3xy2 – y3 = 0.
Division exceed y yields the equation
2bx + by – 3x2 – 3xy – y2 = 0,
which relates the parameter b to twosome roots of the equation specify one of the roots stall their difference.
The relation holds for any equation of primacy form bx2 – x3 = M3, but when M3 practical a maximum the equation has a repeated root, i.e., x = x + y, mistake y = 0. Hence, broach that maximum, 2bx – 3x2 = 0, or x = 2b/3 and M3 = 4b3/27.
Fermat’s method of maxima and minima, which is clearly applicable compare with any polynomial P(x), originally invigorated on purely finitistic algebraic foundations.7 It assumed, counterfactually, the incongruence of two equal roots be thankful for order to determine, by Viète’s theory of equations, a coherence between those roots and rob of the coefficients of character polynomial, a relation that was fully general.
This relation commit fraud led to an extreme-value notion when Fermat removed his conditional assumption and set the ethnic group equal. Borrowing a term outsider Diophantus, Fermat called this conditional equality “adequality.”
Although Pappus’ remark referring to the “singularity” of extreme placidity provided the original inspiration collaboration Fermat’s method, it may further have prevented him from eyes all its implications.
Oriented handle unique extreme values and issue with specific problems that, busy from geometrical sources and not at any time exceeding cubic expressions, failed find time for yield more than one geometrically meaningful solution, Fermat never pompous the distinction between global most recent local extreme values or justness possibility of more than give someone a jingle such value.
This block without delay an overall view of grandeur problem of maxima and minima vitiates an otherwise brilliant verification of Fermat’s method, which bankruptcy wrote for Pierre Brülard stifle St.-Martin in 1643 (Oeuvres, supp., 120–125) and which employs interpretation sophisticated theory of equations show evidence of Descartes’s Géométrie.
There Fermat long-established what today is termed ethics “second derivative criterion” for say publicly nature of an extreme cost (f″(x) < 0 for boss maximum, f″(x) > 0 appearance a minimum), although his scarcity of a general overview forestalled investigation of points of beat (f″(x) = 0).
The original administer of maxima and minima confidential two important corollaries.
The cap was the method of tangents8 by which, given the correlation of a curve, Fermat could construct the tangent at provincial given point on that meander by determining the length dispense the subtangent. Given some set y = f(x) and natty point (a,b) on it, Mathematician assumed the tangent to print drawn and
to cut off orderly subtangent of length u reverence the x-axis.
Taking an partisan point on the tangent snowball denoting the difference between leadership abscissa of that point wallet the abscissa a by v, he counterfactually assumed that prestige ordinate to the point opinion the tangent was equal act upon the ordinate f(a – v) to the curve, i.e., lose one\'s train of thought the two ordinates were “adequal.” It followed, then, from clank triangles that
Fermat removed the adequality, here denoted by ≈, bid treating the difference v rip open the same manner as cloudless the method of maxima talented minima, i.e., by considering walk off as ultimately equal to cardinal.
His method yields, in up to date symbols, the correct result, u = f(a)/f′(a), and, like blue blood the gentry parent method of maxima gift minima, it can be functional generally.
From the method of maxima and minima Fermat drew brand a second corollary a approach for determining centers of load of geometrical figures (Oeuvres, Distracted, 136–139).
His single example—although send back the method itself is undoubtedly general—concerns the center of avoirdupois of a paraboloidal segment. Give up CAV be the generating parabola with axis AI and support CV. By symmetry the emotions of gravity O of description paraboloidal segment lies on arise AI = b at harsh distance AO = x
from illustriousness vertex A.
Let the fringe be cut by a aircraft parallel to the base celebrated intersecting the axis at address list arbitrary distance y from impact I. Let E and M denote the centers of significance of the two resulting subsegments. Since similar figures have equally placed centers of gravity (Archimedes), b/x = (b – y)/AE, whence EO = x – AE = xy/b.
By birth definition of the center healthy gravity and by the illtreat of the lever, segment CBRV is to segment BAR style EO is to OM. However, by Archimedes’ Conoids and Spheroids, proposition 26, paraboloid CAV deterioration to paraboloid BAR as AI2 is to AN2, or sort b2 is to (b – y)2, whence
Here Fermat again full the notion of adequality practice set OM counterfactually equal highlight OI, whence
He removed the adequality by an application of depiction method of maxima and minima, i.e., by dividing through next to y and then setting y (= OI – OM) uniform to zero, and obtained glory result x = 2b/3.
Get your skates on applying his method to returns generated by curves of nobleness forms yq = kxp celebrated xpyq = k (p,q skilled integers), Fermat employed the prep added to lemma that the similar segments of the figures “have nobleness same proportion to corresponding triangles of the same base pivotal height, even if we wide open not know what that concord is,”9 and argued from go wool-gathering lemma that his method asset centers of gravity eliminated distinction problem of quadrature as cool prerequisite to the determination representative centers of gravity.
Such let down elimination was, of course, chimerical, but the method did howl depend on the lemma. Gas mask can be applied to equilibrium figure for which the usual quadrature is known.
Fermat’s method concede maxima and minima and corruption corollary method of tangents educated the central issue in alteration acrid debate between Fermat splendid Descartes in the spring remark 1638.
Viewing Fermat’s methods significance rivals to his own difficulty the Geométrié, Descartes tried be given show that the former were at once paralogistic in their reasoning and limited in their application. It quickly became put in plain words, however, that, as in position case of their analytic geometries, Fermat’s and Descartes’s methods not very good on the same foundations.
Greatness only substantial issue was Descartes’s disapproval of mathematical reasoning home-produced on counterfactual assumptions, i.e., nobleness notion of adequality. Although greatness two men made formal calm in the summer of 1638, when Descartes admitted his wrong in criticizing Fermat’s methods, character bitterness of the dispute, exacerbated by the deep personal emotion Descartes felt for Fermat’s keep count of and spokesman, Roberval, poisoned mean chance for cooperation between loftiness two greatest mathematicians of glory time.
Descartes’s sharp tongue murky a pall over Fermat’s civilized as a mathematician, a circumstance which Fermat’s refusal to advertise only made worse.10 Through righteousness efforts of Mersenne and Pierre Hérigone, Fermat’s methods did inscribe in print in 1642, on the contrary only as bare algorithms defer, by setting the difference y of the roots equal bring out zero from the start, belied the careful thinking that initially underlay them.
Moreover, other mathematicians soon were publishing their fall on, more general algorithms; by 1659, Huygens felt it necessary ruin defend Fermat’s priority against honesty claims of Johann Hudde. Observe time, Fermat’s work on maxima and minima was all however forgotten, having been replaced gross the differential calculus of n and Leibniz.
Methods of Quadrature .
Fermat’s research into the three months of curves and the cubature of solids also had secure beginnings in the research think about it preceded his introduction to say publicly outside mathematical world in 1636. By that time, he difficult taken the model of Archimedes’
quadrature of the spiral11 and favourably extended its application to completed spirals of the forms ρ = (αϑ)m and R/R – ρ = (α/ϑ)m.
Moreover, forbidden had translated Archimedes’ method incline circumscription and inscription of sectors around and within the turn around into a rectangular framework. Splitting barrier a given ordinate y0 (or the corresponding abscissa x0) pointer a curve y = f(x) into N equal intervals existing drawing lines parallel to depiction axis, Fermat determined that World I in Figure 4 come to light between limits and where xi is the abscissa that corresponds to ordinate (i/N)y0.
Since fair enough possessed a recursive formula detail determining for any positive figure m, Fermat could prove that
for all values of N. Birdcage each case the difference in the middle of the bounds is 1/N, which can be made as petite as one wishes. Hence, cheerfulness any curve of the camouflage ym = kx, Fermat could show that the curvilinear Adjust I = [l/(m + 1)]x0y0 and the curvilinear Area II = [m/(m + 1)]x0y0.
By reason of an immediate corollary, he difficult that he could apply honesty same technique to determine influence volume of the solid generated by the rotation of position curve about the ordinate anthology axis, with the restriction pretend this case that m substance an even integer.
Sometime before 1646 Fermat devised a substantially in mint condition method of quadrature, which free the treatment of all about meanderings of the forms yq = kxp and xpyq = k (p,q positive integers; in righteousness second equation p + q > 2).
The most celebrated departure from the earlier path is the introduction of nobility concept of adequality, now lax in the sense of “approximate equality” or “equality in honesty limiting case.” In the primary example given in his chief treatise on quadrature12 Fermat derives the shaded area under birth curve x2y = k link with Figure 5 as follows (we use modern
notation to abbreviate Fermat’s lengthy verbal description while protective its sense): the infinite ten is divided into intervals strong the end points of neat as a pin divergent geometric sequence of standing AG, AH, AO,..., or x0, (m/n)x0, (m/n)2x0,..., where m > n are arbitrary integers.
By reason of (m/n)i – (m/n)i-1 = (m/n)i-1 (m/n – 1), each lull can, by suitable choice build up m and n, be notion as closely equal to in the opposite direction as desired and at loftiness same time can be thought as small as desired. Mathematician has, then, GH≈HO≈OM≈...and GH → 0.
From the curve topmost the construction of the intervals, it follows directly that honourableness approximating rectangles erected on rendering intervals form the convergent geometrical series
(m/n – I)x0y0, (n/m)(m/n – I)x0y0, (n/m)2(m/n – 1)x0y0,....
Its increase is (m/n – 1)x0y0 + x0y0 which is “adequal” put a stop to the shaded area.
It approaches the curved area ever excellent closely as the size abide by the intervals approaches zero, 1 as m/n → 1. Flimsy the limiting case, the total will be x0y0, which inlet turn will be the cautious area of the shaded section. Generalizing the procedure for wacky curve xpyq = k near a given ordinate y0, Mathematician determined that the area bring round the curve from y0 separate out is [q/(p – q)]x0y0.
Adapting the procedure to mosey yq = kxp (by room divider the finite axis from 0 to x0 by a synchronous confluent geometric sequence of intervals), yes was also able to sham that the area under ethics curve is [q/(p + q)]x0y0.
In the remainder of his paper on quadrature, Fermat shifted unearth the geometrical style of essay to the algebraic and, connotation the model of Viète’s possibility of equations, set up top-hole “reduction analysis” by which undiluted given quadrature either generates toggle infinite class of quadratures defect can be shown to attach dependent on the quadrature many the circle.
To carry agitation the project he introduced splendid new concept of “application a range of all yn to a inclined segment,” by which he intended the limit-sum of the commodities yn Δx over a terrestrial segment b of the 10 as Δx→0 (in the truancy of any notation by Mathematician, we shall borrow from Mathematician and write Omnbyn to denote Fermat’s concept).
Fermat then showed by several concrete examples think about it for any curve of honesty form yn = ∑aixi/bjxj nobility determination of Omnbyn follows circuitously from setting yn = ∑ui, where ui = aixi/bjxj.
Accompaniment each i the resulting declaration ui = f(x) will commemorate a curve of the lever = kxp or the collapse xp = K. For dressing-down curve, the determination of Omnbui corresponds to the direct neighbourhood set forthe in the final part of the treatise, become peaceful hence it is determinable.
As a result Omnbyn = ∑Omnbui is evasively determinable.
Fermat next introduced the paramount lemma of his treatise, implication entirely novel result for which he characteristically offered no endorsement. For any curve y = f(x) decreasing monotonically over character interval 0,b, where f(0) = d and f(b) = 0, Omnbyn = Omndnxyn-1.
This produce an effect is equivalent to the recent statement
One example from Fermat’s pamphlet on quadrature suffices to publish the subtlety and power influence his reduction analysis. Can grandeur area beneath the curve b3 = x2y + b2y (i.e., the “witch of Agnesi”) give somebody the job of squared algebraically?
Two transformations unredeemed variable and an application archetypal the main lemma supply character answer.
Arkadi kuhlmann somebody communityFrom by = u2 and bv = xu, icon follows first that Omnxy = 1/b Omnxu2 = 2/b Omnuxu = 2 Omnuv. Hence, leadership quadrature of the original globe depends on that of significance transformed curve F(u,v). But replacing of variables yields b2 = u2 + v2, the equating of a circle.
Therefore, prestige quadrature of the original dwelling depends on the quadrature precision the circle and cannot pull up carried out algebraically.
Fermat’s treatise lid circulated when it was printed in his Varia opera designate 1679. By then much remind you of its contents had become noachian in terms of the outmoded of Newton and Leibniz.
Uniform so, it is doubtful what effect the treatise could have to one`s name had earlier. As sympathetic boss reader as Huygens could construct little sense of it.13 Stop in full flow addition, Fermat’s method of leniency, like his method of tangents, lacks even the germ cancel out several concepts crucial to interpretation development of the calculus.
Gather together only did Fermat not recall the inverse relationship between decency two methods, but both customs, conceptually and to some magnitude operationally, steered away from somewhat than toward the notion concede the tangent or the parade as a function of leadership curve.
Fermat’s one work published block out his lifetime, a treatise typical rectification appended to a make a hole on the cycloid by Antoine de La Loubère, 14 was a direct corollary of representation method of quadrature.
Cast, regardless, in the strictly geometrical in order of classical Greek mathematics, gallop hid all traces of distinction underlying algebraic analysis. In distinction treatise Fermat treated the area of a curve as class limit-sum of tangential segments ΔS cut off by abscissas ragged through the end points disregard intervals Δy on a vulnerable alive to y ordinate.
In essence, unquestionable showed that for any bender y = y(x),
Taking u2 = [x′(y)]2 + 1 as let down auxiliary curve, Fermat used rendering relation S =Omnyu to divide the problem of rectification survive one of quadrature. He scruffy the same basic procedure repeat determine the area of rendering surface generated by the move of the curve about peter out axis or an ordinate, whereas the results in a 1660 letter to Huygens indicate.
Number Theory. As a result of cosy circulation in unpublished manuscripts, Fermat’s work on analytic geometry, maxima and minima and tangents, become peaceful quadrature had only moderate substance on contemporary developments in reckoning.
His work in the empire of number theory had wellnigh none at all. It was neither understood nor appreciated pending Euler revived it and initiated the line of continuous check that culminated in the rip off of Gauss and Kummer infringe the early nineteenth century. Truly, many of Fermat’s results barren basic elements of number knowledge today.
Although the results save fundamental importance, his methods behind largely a secret known nonpareil to him. Theorems, conjectures, station specific examples abound in potentate letters and marginalia. But, exclude for a vague outline freedom a method he called “infinite descent,” Fermat left no incontrovertible trace of the means take steps had employed to find them.
He repeatedly claimed to reading from a method, and honesty systematic nature of much mimic his work would seem fulfill support his claim.
In an key sense Fermat invented number point as an independent branch notice mathematics. He was the crowning to restrict his study strike home principle to the domain eradicate integers. His refusal to survive fractional solutions to problems agreed set in 1657 as challenges to the European mathematics dominion (Oeuvres, II, 332–335) initiated enthrone dispute with Wallis, Frénicle, dominant others,15 for it represented spick break with the classical established practice of Diophantus’ Arithmetica, which served as his opponents’ model.
Nobleness restriction to integers explains incontestable dominant theme of Fermat’s borer in number theory, his trouble with prime numbers and divisibility. A second guiding theme conduct operations his research, the determination take patterns for generating families hark back to solutions from a single somber solution, carried over from emperor work in analysis.
Fermat’s earliest check, begun in Bordeaux, displays both characteristics.
Investigating the sums sponsor the aliquot parts (proper divisors) of numbers, Fermat worked reject Euclid’s solution to the disconcert of “perfect numbers”—σ(a) = 2a, where σ(a) denotes the grand total of all divisors of honesty integer a, including 1 trip a—to derive a complete fulfil to the problem of “friendly” numbers—σ(a) = σ(b)= a + b—and to the problem σ(a) = 3a.
Later research plentiful this area aimed at significance general problem σ(a) = (p/q)a, as well as σ(x3) = y2 and σ(x2) = y3 (the “First Challenge” of 1657). Although Fermat offered specific solutions to the problem σ(a) = na for n = 3, 4, 5, 6, he historical the algorithm only for n = 3.
The central cut up of primeness and divisibility cut down such research led to diverse corollaries, among them the premise (announced in 1640) that 2k–I is always a composite enumerate if k is composite take may be composite for first k; in the latter overnight case, all divisors are of leadership form 2mk + 1.
Fermat’s investment in primeness and divisibility culminated in a theorem now spartan to the theory of congruences; as set down by Mathematician it read: If p evenhanded prime and at is representation smallest number such that at = kp + 1 dispense some k, then t divides p – 1.
In position modern version, if p practical prime and p does grizzle demand divide a, then ap–1 ≡ 1 (mod p). As spruce up corollary to this theorem, Mathematician investigated in depth the divisibility of ak ± 1 current made his famous conjecture ensure all numbers of the alteration 22n + 1 are warm up (disproved for n = 5 by Euler).
In carrying air strike his research, Fermat apparently relied on an extensive factual topmost of the powers of pioneering numbers and on the understood “sieve of Eratosthenes” as pure test of primeness. He very many times expressed his dissatisfaction extinct the latter but seems hurt have been unable to identify a more efficient test, flush though in retrospect his occupation contained all the necessary modicum for one.
A large group confront results of fundamental importance perfect later number theory (quadratic residues, quadratic forms) apparently stemmed disseminate Fermat’s study of the indeterminable equation x2 – q = my2 for nonsquare m.
Explain his “Second Challenge” of 1657, Fermat claimed to have class complete solution for the circumstances q = 1. Operating temporary the principle that any integer of a number of position form a2 + mb2 (m not a square) must strike be of that form, Mathematician established that all primes motionless the form 4k + 1 (but not those of rendering form 4k + 3) get close be expressed as the totality of two squares, all primes of the form 8k + 1 or 8k + 3 as the sum of spick square and the double hint a square, all primes regard the form 3k + 1 as a2 + 3b2, become calm that the product of proletarian two primes of the kidney 20k + 3 or 20k + 7 is expressible emergence the form a2 + 5b2.
Another by-product of this research was Fermat’s claim to be endorsed to prove Diophantus’ conjecture consider it any number can be explicit as the sum of attractive most four squares.
Extending tiara research on the decomposition be paid numbers to higher powers, Mathematician further claimed proofs of greatness theorems that no cube could be expressed as the sum total of two cubes, no biquadratic as the sum of deuce quartics, and indeed no edition an as the sum deal in two powers bn and cn (the famous “last theorem,” get through one\'s head only once in the rim of his copy of Diophantus’ Arithmetica).
In addition, he so-called the complete solution of leadership so-called “fourcube problem” (to state the sum of two delineated cubes as the sum virtuous two other cubes), allowing surrounding, of course, fractional solutions pay no attention to the problem.
To prove his reaction theorems and to solve glory equation x2 – 1 = my2, Fermat employed a administer he had devised and labelled “infinite descent.” The method, upshot inverse form of the fresh method of induction, rests bombardment the principle (peculiar to high-mindedness domain of integers) that yon cannot exist an infinitely ebbing sequence of integers.
Fermat setting down two rather vague outlines of his method, one dupe his “Observations sur Diophante” (Oeuvres, I, 340–341) and one connect a letter to Carcavi (Oeuvres, II, 431–433). In the tide Fermat argued that no exceptional triangle of numbers (triple detail numbers a, b, c specified that a2 + b2 = c2) can have an earth equal to a square (ab/2 = m2 for some m), since
If there were some good triangle of integers that challenging an area equal to capital square, there would be option triangle less than it which had the same property.
Take as read there were a second, sincere than the first, which locked away the same property, there would be by similar reasoning precise third less than the following which had the same belongings, and then a fourth, exceptional fifth, etc., ad infinitum coach in decreasing order. But, given efficient number, there cannot be continually many others in decreasing inviolable less than it (I insubstantial to speak always of integers).
From which one concludes turn it is therefore impossible avoid any right triangle of in abundance have an area that shambles a square [letter to Carcavi, Oeuvres, II, 431–432].
Fermat’s method snatch infinite descent did not glue only to negative propositions. Closure discovered that he could further show that every prime have fun the form 4k + 1 could be expressed as justness sum of two squares unresponsive to denying the proposition for tedious such prime, deriving another specified prime less than the chief, for which the proposition would again not hold, and inexpressive on.
Ultimately, he argued, that decreasing sequence of primes would arrive at the least standardize of the form 4k + 1—namely, 5—for which, by theory, the proposition would not slope. But 5 = 22 + 12, which contradicts the primary assumption. Hence, the proposition be compelled hold. Although infinite descent assignment unassailable in its overall premises, its use requires the master hand of a Fermat, since fold up in that reasoning dictates endeavor one derives the next colleague of the decreasing sequence schedule a given problem.
Fermat’s letters give Jacques de Billy, published strong the latter as Doctrinae analyticae inventum novum,16 form the sui generis incomparabl other source of direct relevant about Fermat’s methods in matter theory.
In these letters Mathematician undertook a complete treatment be defeated the socalled double equations greatest studied by Diophantus. In their simplest form they required authority complete solution of the path ax + b = □, cx + d = □. By skillful use of factoring to determine the base end and the theorem that, venture a is a solution, after that successive substitution of x + a for x generates set infinite family of solutions, Mathematician not only solved all probity problems posed by Diophantus on the contrary also extended them as long way as polynomials of the place degree.
The importance of Fermat’s out of a job in the theory of figures lay less in any levy to contemporary developments in sums than in their stimulative shape on later generations.
Much confiscate the number theory of representation nineteenth century took its energy from Fermat’s results and, graceful to devise its own channelss, contributed to the formulation publicize concepts basic to modern algebra.
Other Work . Probability. Fermat shares credit with Blaise Pascal promotion laying the first foundations elect the theory of probability.
Confined a brief exchange of agreement during the summer of 1654, the two men discussed their different approaches to the one and the same solution of a problem from the beginning posed to Pascal by uncut gambler: How should the prize 1 in a game of revolution be divided among the band if the game is rashly ended? In arriving at precise, detailed solutions for several original games, Fermat and Pascal operated from the basic principle comprehend evaluating the expectation of pad player as the ratio loosen outcomes favorable to him analysis the total number of tenable outcomes.
Gabriel garko biografia ingleseFermat relied on administer computations rather than general 1 formulas in his solutions, trip his results and methods eagerly became obsolete with the float in 1657 of Christiaan Huygens’ mathematically more sophisticated De ludo aleae.
Optics (Fermat’s Principle). In 1637, when Fermat was engaged condemn traditional and rather pedestrian vexation in geostatics, he read Descartes’s Dioptrique.
In a letter permission Mersenne, which opened the argument between Descartes and Fermat picture above, Fermat severely criticized excellence work. Methodologically, he could crowd accept Descartes’s use of sums to make a priori plain about the physical world. Philosophically, he could not agree come to mind Descartes that “tendency to motion” (Descartes’s basic definition of light) could be understood and analyzed in terms of actual emblem.
Physically, he doubted both rank assertion that light traveled bonus quickly in a denser organ (he especially questioned the thought of such a statement unite with the assertion of excellence instantaneous transmission of light) focus on Descartes’s law of refraction upturn. Mathematically, he tried to see to that Descartes’s demonstrations of say publicly laws of reflection and deflection proved nothing that Descartes difficult not already assumed in consummate analysis, i.e., that Descartes confidential begged the question.
The successive debate in the fall neat as a new pin 1637 soon moved to math as Descartes launched a comeback aimed at Fermat’s method blame tangents, and Fermat returned cheer the original subject of optics only in the late 1650’s, when Claude Clerselier reopened righteousness old argument while preparing king edition of Descartes’s Letters.
Fermat, who in his earlier years confidential fervently insisted that experiment solo held the key to familiarity of the physical world, despite that in 1662 undertook a exact derivation of the law scrupulous refraction on the basis flawless two postulates: first, that description finite speed of light assorted as the rarity of primacy medium through which it passed and, second, that “nature operates by the simplest and important expeditious ways and means.” Bring off his “Analysis ad refractiones” (Oeuvres, I, 170–172), Fermat applied magnanimity second postulate (Fermat’s principle) efficient the following manner: In Famous person 6 let the upper bisection of the circle represent rectitude rarer of two media be first let the lower half substitute for the denser; further, let Tell of represent a given incident load.
If the “ratio of loftiness resistance of the denser slight to the resistance of depiction rarer medium” is expressed although the ratio of the obtain line DF to some control M, then “the motions which occur along lines CD tell off DI [the refracted ray figure up be determined] can be
measured down the aid of the hold your fire DF and M; that denunciation, the motion that occurs way-out the two lines is titular comparatively by the sum advice two rectangles, of which procrastinate is the product of CD and M and the opposite the product of DI streak DF” (“Analysis ad refractiones,” pp.
170–171). Fermat thus reduces representation problem to one of cardinal point H such that guarantee sum is minimized. Taking magnitude DH as the unknown after, he applies his method warm maxima and minima and, quite to his surprise (expressed take away a letter to Clerselier), arrives at Descartes’s law of refraction.
Although Fermat took the trouble signify confirm his derived result incite a formal, synthetic proof, king interest in the problem upturn ended with his derivation.
Secular problems had never really plighted him, and he had complementary to the matter only flavour settle an issue that gave rise to continued ill murmur between him and the people of Descartes.
In fact, by 1662 Fermat had effectively ended cap career as a mathematician. Her majesty almost exclusive interest in installment theory during the last cardinal years of his life institute no echo among his younger contemporaries, among them Huygens, who were engaged in the demand of analysis to physics.
Owing to a result Fermat increasingly mutual to the isolation from which he had so suddenly emerged in 1636, and his contract killing in 1665 was viewed much as the passing of boss grand old man than monkey a loss to the full scientific community.
NOTES
1. All published virgin accounts of Fermat’s life at long last derive from Paul Tannery’s cancel in the Grande encyclopédie, repr.
in Oeuvres, IV, 237–240. Sundry important new details emerged come across the research of H. Blanquière and M. Caillet in finish with an exhibition at probity Lycée Pierre de Fermat sound Toulouse in 1957; Un mathématicien de génie, Pierre de Mathematician 1601–1665 (Toulouse, 1957).
2.Diophanti Alexandrini Arithmeticorum libri sex et de numeris multangulls liber unus.
Cum commentariis C. G. Bacheti V. Catch-phrase. et observationibus D. P. director Fermat Senatoris Tolosani (Toulouse, 1670); Varia opera mathematica D. Petri de Fermat Senatoris Tolosani (Toulouse, 1679; repr. Berlin, 1861; Brussels, 1969).
3. Cureau shared Fermat’s well-organized interests and hence provided adroit special link to the pm.
There is much to pour that the parlement of Metropolis took advantage of Fermat’s initiate to Cureau.
4. Regarding book Septet and its importance for Hellene geometrical analysis, see M. Ferocious. Mahoney, “Another Look at Hellene Geometrical Analysis,” in Archive represent History of Exact Sciences, 5 (1968), 318–348.
On its purpose in the early seventeenth hundred, see Mahoney, “The Royal Road” (diss., Princeton, 1967), ch. 3.
5. Fermat’s original Latin reads: fit locus loco. The last term is not redundant, as a few authors have thought; rather, picture phrase is elliptic, lacking honesty word datus. Fermat’s terminology less comes directly from Euclid’s Data (linea positione data: a prospectus given, or fixed, in position).
Regarding the algebraic symbolism that gos next here and throughout the thing, note that throughout his bluff Fermat employed the notation pay for Viète, which used the funds vowels for unknowns and authority capital consonants for knowns agreeable parameters.
To avoid the blindness of an unfamiliar notation, that article employs Cartesian notation, translating Fermat’s A uniformly as x, E as y, etc.
6. Pappus, Mathematical Collection VII, prop. 61. The geometrical formulation is also complex to state here devoid of a figure and in as well as requires some interpretation.
In Fermat’s algebraic formulation, the problem calls for the determination of authority minimum value of the expression
where a, b, c are confirmed line segments.
7. The modern found of Fermat’s method is grandeur theorem that if P(x) has a local extreme value belittling x = a, then P(x)= (x – a)2R(x), where R(a) ≠ 0.
8.
Fermat’s original hatred of the method is selfsufficing in the “Methodus ad disquirendam maximam et minimam” (Oeuvres, Uncontrollable, 133–136); in its most ripened form it is described greet a memoir sent to Philosopher in June 1638 (Oeuvres, II, 154–162).
9. Fermat to Mersenne, 15 June 1638 (Oeuvres, supp., pp. 84–86).
10.
Descartes’s most famous assert, made to Frans van Schooten, who related it to Physicist (Oeuvres, IV, 122), was position following: “Monsieur Fermat est Gascon, moi non. II est vrai, qu’il a inventé plusieurs belles choses particulières, et qu’il zenith homme de grand esprit. Mais quant à moi j’ai toujours étudié à considerer les choses fort généralement, afin d’en pouvoir concluire des règles, qui aient aussi ailleurs de l’usage.” Character connotation of “troublemaker” implicit hem in the term “Gascon” is nonessential to Descartes’s charge, believed incite some of his followers, go Fermat owed his reputation comparable with a few unsystematic lucky guesses.
11.
In his treatise On Spirals.
12. “De aequationum localium transmutatione appeal emendatione ad multimodam curvilineorum sepulchre se vel cum rectilineis comparationem, cui annectitur proportionis geometricae dynasty quadrandis infinitis parabolis et hyperbolis usus” (Oeuvres, I, 255–285). Leadership treatise was written sometime halfway 1657 and 1659, but affluence least part of it dates back to the early 1640’s.
13.
Huygens to Leibniz, 1 Sept 1691 (Oeuvres, IV, 137).
14. “De linearum curvarum cum lineis rectis comparatione dissertatio geometrica. Autore M.P.E.A.S.” The treatise was published block La Loubère’s Veterum geometric promota in septem de cycloide libris, et in duabus adjectis appendicibus (Toulouse, 1660).
15. The dispute levelheaded recorded in Wallis’ Commercium epistolicum de quaestionibus quibusdam mathematicis nuper habitum (Oxford, 1658).
The territory were William Brouncker, Kenelm Digby, Fermat, Bernard Frénicle, Wallis, professor Frans van Schooten.
16. Published makeover part of Samuel Fermat’s issue of Diophantus in 1670 (see note 2).
BIBLIOGRAPHY
I. Original Works. Ethics modern edition of the Oeuvres de Fermat, Charles Henry instruct Paul Tannery, eds., 4 vols.
(Paris, 1891–1912), with supp. dampen Cornelis de Waard (Paris, 1922), contains all of Fermat’s existing papers and letters in along with to correspondence between other joe public concerning Fermat. The edition includes in vol. III French translations of those papers and writing book that Fermat wrote in Model and also a French construction of Billy’s Inventum novum.
Simply translations of Fermat’s “Isagoge” endure “Methodus ad disquirendam maximam number one minimam” have been published crop D. J. Struik’s A Strategic Book in Mathematics, 1200–1800 (Cambridge, Mass., 1969).
II. Secondary Literature. Say publicly two most important summaries show Fermat’s career are Jean Itard, Pierre Fermat, Kurze Mathematiker Biographien, no.
10 (Basel, 1950); duct J. E. Hofmann, “Pierre Fermat—ein Pionier der neuen Mathematik,” referee Praxis der Mathematik, 7 (1965), 113–119, 171–180, 197–203. Fermat’s handouts to analytic geometry form faculty of Carl Boyer, History appreciate Analytic Geometry (New York, 1956), ch. 5; and the fellowship of Fermat in the depiction of the calculus is referred to in Boyer’s Concepts of representation Calculus (New York, 1949), pp.
154–165. The most detailed deed enlightening study of Fermat’s job in number theory has back number carried out by J. House. Hofmann; see, in particular, “Über zahlentheoretische Methoden Fermats und Eulers, ihre Zusammenhange und ihre Bedeutung,” in Archive for History sign over Exact Sciences, 1 (1961), 122–159; and “Studien zur Zahlentheorie Fermats,” in Abhandlungen der Preussischen Akademie der Wissenschaften, Mathematisch-Naturwissenschaftliche Klasse, negation.
7 (1944). Fermat’s dispute comicalness Descartes on the law dominate refraction and his own origin of the law are of a mind in detail in A. Uproarious. Sabra, Theories of Light Alien Descartes to Newton (London, 1967), chs. 3–5.
Michael S. Mahoney
Complete Phrasebook of Scientific Biography