mathematicians-from-belgium

"We present three recent developments in wavelets and subdivision: wavelet-type transforms that map integers to integers, with an application to lossless coding for images; rate-distortion bounds that realize the compression given by nonlinear approximation theorems for a model where wavelet compression outperforms the Karhunen-Loeve approach; and smoothness results for irregularly spaced subdivision schemes, related to wavelet compression for irregularly spaced data."

"Mathematicians have various ways of judging the merits of new theorems and constructions. One very important criterion is esthetic — some developments just “feel” right, fitting, and beautiful. Just as in other venues where beauty or esthetics are discussed, taste plays an important role in this, but I think I am not alone in being especially excited when apparently different fields suddenly meet in a new concept, a new understanding. It is often of the sparks of such encounters that our esthetic enjoyment of mathematics is born. Another important criterion for according merit to some particular piece of mathematics is the extent to which it can be useful in applications; this is the criterion almost exclusively used by nonmathematicians."

"The wavelet transform is a tool that cuts up data or functions or operators into different frequency components, and then studies each component with a resolution matched to its scale. Forerunners of this technique were invented independently in pure mathematics (Calderón's resolution of the identity in harmonic analysis—see e.g., Calderón (1964), physics (coherent states for the (ax + b)-group in quantum mechanics, first constructed by Aslaksen and Klauder (1968), and linked to the hydrogen atom Hamiltonian by Paul (1985)) and engineering (QMF filters by Esteban and Galland (1977), and later QMF filters with exact reconstruction property by Smith and Barnwell (1986), Vetterli (1986) in electrical engineering; wavelets were proposed for the analysis of seismic data by J. Morlet (1983)). The last five years have seen a synthesis between all these different approaches, which has been very fertile for all the fields concerned."

"The development of wavelets is an example where ideas from many different fields combined to merge into a whole that is more than the sum of its parts. The subject area of wavelets, developed mostly over the last 15 years, is connected to older ideas in many other fields, including pure and applied mathematics, physics, computer science, and engineering."

"... In their mathematical aspect, wavelets are rooted in the use of dilations and convolutions in Calderón-Zygmund theory in harmonic analysis. ... Algorithmically, wavelets are related to subband filtering in electrical engineering. Subband filtering was developed from the 70-s on; exact reconstruction procedures were discovered in the early 80-s. These were obviously fast algorithms, meant as a front-end processing step before encoding or compressing information in various types of signals. A lot of effort went into optimizing the filters for various applications, and this subfield of electrical engineering is now quite mature. ... Another algorithmic ancestor of wavelets are the multiple algorithms in numerical analysis, closer to mathematics, but still ad hoc."

"[T]here were... no mathematical methods for determination of pressure distribution over bodies immersed in fluids. Stevinus was... the first... to attempt such... on the basis of his 'solidification principle'. ...[H]e ...established that pressure is independent of the configuration of the body, and depends only on the weight of the column of water above it. The other original contribution... was the establishment of the... 'hydrostatic paradox': the pressure experienced by the bottom of the vessel containing a fluid depends only on the (horizontal) area of the bottom and the depth below the surface of the fluid, but... not... on the shape of the vessel. ...Stevinus described how he and ...Gretius had experimented ...and found that a lightweight and heavyweight body dropped from the same height took the same time to reach the ground. This was contrary to Aristotle's theory..."

"The works on which the fame of Diophantus rests are: 1. the Arithmetica (originally in thirteen Books) 2. a tract On Polygonal Numbers. Six Books only of the former and a fragment of the latter survive. ...In 1585 Simon Stevin published a French version of the first four Books, based on Xylander."

"Long before Mouton's proposals of 1670... Simon Stevin... published, first in Flemish and in the same year, 1585, in French, a treatise in which the first explanation of decimal fractions is given. In the same treatise Stevin proposes that not only weights and moneys but also linear, square, and cubic measure and even degrees and minutes should be reduced to a decimal system. This proposal, together with the explanation of the decimal fractions, establishes for Simon Stevin a proud place in the history of the development of scientific systems of measurement."

"Among the ancients, Archimedes was the only one who attained clear and correct notions on theoretical statics. He had acquired firm possession of the idea of pressure, which lies at the root of mechanical science. But his ideas slept nearly twenty centuries until the time of S. Stevin and Galileo Galilei. Stevin determined accurately the force necessary to sustain a body on a plane inclined at any angle to the horizon. He was in possession of a complete doctrine of equilibrium. While Stevin investigated statics, Galileo pursued principally dynamics."

"...the use of the Disme ...to teach such as doe not already know the use and practize of Numeration, and the foure principles of common Arithmetick, in whole numbers, namely, Addition, Substraction, Multiplication, & Division, together with the Golden Rule, sufficient to instruct the most ignorant in the usuall practize of this Art of Disme or Decimall Arithmeticke"

"I shall lead the industrious learner a few steps farther in order to his understanding the resolution of all manner of compound equations in numbers, and... shall explain Simon Stevin's General Rule, which, with the help of the rules in the following eleventh Chapter, will discover all the roots of any possible equation in numbers, either exactly, if they be rational, or very nearly true, if irrational."

"The first systematic discussion of decimal fractions with full appreciation of their significance was given by Simon Stevin... His work in Flemish, entitled La Thiende... was republished again in 1585 in French with the title La Disme; in 1608 an English translation by Robert Norton, The Art of Tenths or Decimall Arithmetike, appeared in London. This work is addressed to astronomers, surveyors, masters of money (of the mint), and to all merchants. ...All Stevin says applies today, hardly with a change of letter. ...The immediate application of decimal fractions was made particularly to the trigonometric functions and to logarithms"

"The first conception of a changing conception of number are found in Regiomontanus. He was a man who still stood with one foot in the world of the Ancients, and hence considered numbers as arithmoi, or sets of units. However... the other foot... stood firmly in the modern world, as all magnitudes are quantities 'that are measured in relation to a certain unit'. According to Regiomontanus... 'it is better to approximate the truth, than to ignore it.' With Stevin, this reluctance disappears altogether. His definition of number builds on Regiomontanus, but it was also revolutionary, for he dropped the classical definition altogether and accepted the modern notion: "Nombre est cela par lequel s'explique la quantité de chascune chose," [Number is that by which the quantity of each thing is revealed,] "nombre n'est poinct quantité discontinue" [number is not at all disontinuous quantity] and "que l'unité est nombre" [the unit is a number]. Stevin did not consider numbers as a discontinuous spectrum, but as a continuum"

"Stevin (1585): 3② + 4 egales à 2① + 4. Modern form: 3 x^2 + 4 = 2x + 4."

"One of the greatest curiosities of the history of science that Napier constructed logarithms before exponents were used. To be sure, Stifel and Stevin made some attempts to denote powers by indices, but this notation was not generally known,—not even to T. Harriot, whose algebra appeared long after Napier's death. That logarithms flow naturally from the exponential symbol was not observed until much later. ...While F. Vieta represented A^3 by "A cubus" and Stevin x^3 by a figure 3 within a small circle [around it], Descartes wrote a^3."

"Our intention in this Disme is to worke all by whole numbers: for seing that in any affayres, men reckon not of the thousandth part of a mite, grayne, &c. as the like is also used of the principall Geometricians, and Astronomers, in computacions of great consequence, as Ptolome & Johannes Monta-regio have not described their Tables of Arches, Chords, or Sines, in extreme perfection (as possibly they might have done by Multinomiall numbers,) because that imperfection (considering the scope and end of those Tables) is more convenient then such perfection."

"We call the wise age that in which men had a wonderful knowledge of science which we recognize without fail by certain signs, although without knowing who they were, or in what place, or when. ...It has become a matter of common usage to call the barbarous age that time which extends from about 900 or a thousand years up to about 150 years past, since men were for 700 or 800 years in the condition of imbeciles without the practice of letters or sciences—which condition had its origin in the burning of books through troubles, wars, and destructions; afterwards affairs could, with a great deal of labor, be restored, or almost restored, to their former state; but although the afore-mentioned preceding times could call themselves a wise age in respect to the barbarous age just mentioned, nevertheless we have not consented to this definition of such a wise age, since both taken together are nothing but a true barbarous age in comparison to that unknown time at which we state that it [i.e., the wise age] was, without any doubt, in existence."

"The first Part. Of the Definitions of the Dismes. The first Definition. Disme is a kind of Arithmeticke, invented by the tenth progression, consisting in Characters of Cyphers; whereby a certaine number is described, and by which also all accounts which happen in humane affayres, are dispatched by whole numbers, without fractions or broken numbers."

"The sixt Definition. A Whole number is either a unitie, or a compounded multitude of unities."

"The Rule of Three, or Golden Rule of Arithmeticall whole Numbers. Be the three termes given 2 3 4. ...To finde their fourth proporcionall Terme: that is to say, in such Reason to the third terme 4, as the second terme 3, is to the first terme 2 [Modern notation: \frac{x}{4} = \frac{3}{2}]. ...Multiply the second terme 3, by the third terme 4, & giveth the product 12: which dividing by the first terme 2, giveth the Quotient 6: I say that 6 is the fourth proportional terme required."

"If all this be not put in practize... it wil be beneficiall to our successors, if future men shal hereafter be of such nature as our predecessors, who were never negligent of so great advantage. ...they may all deliver them selves when they will, from so much and so great labour."

"To Simon Stevin of Bruges in Belgium, a man who did a great deal of work in most diverse fields of science, we owe the first systematic treatment of decimal fractions. In his La Disme (1585) he describes in very express terms the advantages, not only of decimal fractions, but also of the decimal division in systems of weights and measures. Stevin applied the new fractions "to all the operations of ordinary arithmetic." What he lacked was a suitable notation. ...Stevin found the greatest common divisor of x^3 + x^2 and x^2 + 7x + 6 by the process of continual division, thereby applying to polynomials Euclid's mode of finding the greatest common divisor of numbers, as explained in Book VII of his Elements. Stevin was enthusiastic not only over decimal fractions, but also over the decimal division of weights and measures. He considered it the duty of governments to establish the latter. He advocated the decimal subdivision of the degree. No improvement was made in the notation of decimals till the beginning of the seventeenth century."

"In the... fifteenth century, the sexagesimal division of the radius, in terms of which cords and goniametrical line-segments were expressed, was generally superseded, though not immediately replaced, by a decimal system of positional notation. Instead, mathematicians sought to avoid fractions by taking the Radius equal to a number of units of length of the form 10^n...The first to apply this method was the German astronomer Regiomontanus... the second half of the sixteenth and the first decades of the seventeenth century... observed of a gradual development of this method of Regiomontanus into a complete system of decimal positional fractions. Yet none of the steps taken by... writers is comparable in importance and scope with the progress achieved by Stevin in his De Thiende."

"The idea of extending the decimal place-value system to include fractions was discovered by several mathematicians. The most influential... was Simon Stevin... who popularized the system in a booklet called De Thiende (“The tenth”), first published in 1585. By extending place value to tenths, hundredths, and so on, Stevin created the system... More importantly, he explained how it simplified calculations... Stevin was aware that his system provided a way to attach a "number" (...decimal expansion) to every... length. ...In his Arithmetic he declared that... roots were just numbers. ...that "there are no absurd, irrational, irregular, inexplicable, or surd numbers" ...all terms for irrational numbers ...Stevin was proposing ...to flatten the incredible diversity of "quantities" or "magnitudes" into one expansive notion of number, defined by decimal expansions. ...this amounted to a fairly clear notion of... the positive real numbers. Stevin's proposal was made immensely more influential by the invention of logarithms. Like the sine and the cosine, these were practical computational tools. ...they needed to be tabulated, and the tables were given in decimal form. Very soon, everyone was using decimal representation. ...The positive real numbers are... an immensely larger number system, whose internal complexity we still do not fully understand."

"Stevin, Stevinus, (Simon) a Flemish mathematician of Bruges, who died in 1633. He was master of mathematics to prince Maurice of Nassau, and inspector of the dykes in Holland. It is said he was the inventor of the sailing chariots, sometimes made use of in Holland. He was a good practical mathematician and mechanist, and was author of several useful works: as treatises on Arithmetic, Algebra, Geometry, Statics, Optics, Trigonometry, Geography, Astronomy, Fortification, and many others, in the Dutch language which were translated into Latin, by Snellius and printed in 2 volumes folio. There are also two editions in the French language, in folio, both printed at Leyden, the one in 1608, and the other in 1634, with curious notes and additions by ."

"Sailing carriages are said to have been used in very remote times in China, and as far back as 1617, in the collection of travels by ... But before that date sailing chariots had actually been constructed in Holland by the celebrated mathematician Simon Stevin (born at Bruges, 1548, died 1620), a much-esteemed friend of the Stadtholder, Prince Maurice of Nassau. These vehicles attracted consider able attention from the men of science of the seventeenth century. Our own Bishop Wilkins is loud in their praise, and Grotius wrote several poems on the carriages and on their constructor. Fortunately, too, there is an engraving, now extremely rare, by Swanenburch, after a design by Jacques de Gheyn, which brings out the arrangement very clearly. It is dated 1612... Another issue of the plates, supposed to be the third, is dated 1652, and a reduced and reversed reproduction is stated by Müller, De Nederlandsche Geschiedenis in Platen, to be found in Bleau's Tooneel der Steden, 1649. A reduced copy of the central plate, showing the carriage only, is given in Le Magazin Pittoresque for 1844, and it is from that copy that the accompanying illustration (Fig. 1) has been produced. ... In a pamphlet, possibly issued with the engraving, bearing the title Windt-Wagens: Les Artificiels Chariots à Voiles du Compte Maurice... is given an account of a journey made apparently in the year 1600 along the beach from the now fashionable Dutch watering-place Scheveningen to Petten, a distance of forty-two miles to the north, which was covered in two hours, a speed which seems almost incredible. The passengers included Prince Maurice himself, who steered; Grotius, then a lad of fifteen; the Spanish Admiral, Francis Mendoza, at that time a prisoner in the hands of Prince Maurice after the battle of Nieuport; and others to the number of twenty-eight. The trial appears to have been a great success, but in spite of this, unless, indeed, the trip performed by De Peiresc in 1606 was made in it, there appears to be absolutely no record of its having been afterwards used, and, stranger still, it is quite unknown what became of the carriage in the end. It is referred to in Howell's Letters as being one of two wonderful things to be seen near the Hague: "A waggon, or ship, or a monster mixed of both, like the Hippocentaur, who was half man and half horse; this engine that hath wheels and sails, will hold above twenty people, and goes with the wind, being drawn or mov'd by nothing else, and will run, the wind being good, and the sails hois'd up, above fifteen miles an hour upon the even hard sands: they say this invention was found out to entertain Spinola when he came hither to treat of the last truce." The anonymous author of The Present State of Holland, 1765, says of Scheveningen: " This village is famous also for a sailing chariot belonging to Prince Maurice, and kept here." He adds: "The last time it made its appearance on the strand was about 17 years ago, when through the unskilfulness of the steersman it had like to have run into the sea, and put the passengers into no small fright." This in all probability refers to the smaller carriage mentioned above as represented in the drawing of Jacques de Gheyn, which was to be seen at Scheveningen as late as 1802; its fate since that date is unknown. There is an account of another, but partially successful trial, with this carriage in 1790 upon the occasion of a royal marriage, and it is known that it was sold by auction in 1795. Stevin's contrivances appear to have set the anonymous author referred to above at work upon his own account, on what must be regarded as the forerunner of the motor perambulators, with which we shall no doubt become familiar ere long."

"For Stevin, the "signs" that in earlier times a "golden age" (aurea aetas) of science actually existed are these: 1. The traces of a perfected astronomical knowledge found in Hipparchus and Ptolemy, whose writings he understands as mere "vestiges" of primeval knowledge... 2. Algebra, as we have become acquainted with it through Arabic books and which represents one of the strangest "vestiges" of the "wise age." No trace of it is found in the Chaldeans, the Hebrews, the Romans, and even the Greeks... 3. Evidence of the foreign origin Greek geometry. ... 4. Information concerning the height of the clouds, which appears in an Arabic work and which Stevin does not hesitate to trace back to the science of the "wise age." "Alchemy," which was unknown to the Greeks and whose most expert representative Stevin saw as Hermes Trismegistos."

"The Dutch engineer Simon Stevin learned about the pressure exerted by water on the walls of canals, and made precise observations of the nature of stable and unstable equilibrium of bodies. He also studied the motion of bodies on slopes."

"Trigonometrical solutions may... be extended to quadrilateral and other multilateral figures, plane as well as spherical, as may be seen in Simon Stevin."

"Some will have it that the Balance, and Steel-yard derive their Origin and fundamental Principles, from these two general Axioms in Mechanicks (viz.) that Equal Weights weigh equally at equal Distances, but unequally, it unequal Distances: and this other, that unequal Weights weigh unequally at equal Distances, but that they weigh equally at unequal Distances, provided that their Distances are in a reciprocal Proportion to their Weights. Those who would be satisfied as to these Demonstrations, may find them In Guido Ubaldus, Galileus, Simon Stevin, John Buteo, in Guevara, and several other Mechanical Writers, who have enlarged very much upon this Subject."

"Positional numeration had been in full use for many centuries before it was realized that among the advantages of the method was its great facility in handling fractions. Even then the realization was far from complete, as may be gleaned from the cumbersome superscripts and subscripts used by Stevin and Napier. ...all that was necessary to bring the scheme to full effectiveness was a mark such as our modern decimal point... Yet... the innovators... with the exception of Kepler and Briggs, either did not recognize this fact, or else had no faith that they could induce the public to accept it. Indeed, a century after Stevin's discovery, a historian... remarked Quod homines tot sententiae (As many opinions as there are people), and it took another century before decimal notation was finally stabilized and the superfluous symbols dropped. [Included table indicates Simon Stevin's notation for 24.375 was: 24 3^{(1)} 7^{(2)} 5^{(3)}.]"

"[The books of Euclid pass on to us] something admirable and very necessary to see and to read, namely the order in the method of writing on mathematics in that aforementioned time of the wise age."

"Simon Stevin... wrote in Latin a book on mathematics, which was published in Leijden in 1608, in which he includes several chapters on bookkeeping. These were a reproduction of a book published in the Dutch language on "bookkeeping for merchants and for princely governments," which appeared in Amsterdam in 1604, and was rewritten in The Hague in 1607, in the form of a letter addressed to Maximiliaen de Bethune, Duke of Seulley. This Duke was superintendent of finance of France and had numerous other imposing titles. He had been very successful in rehabilitating the finances of France and Stevin, knowing him through Prince Maurits of Orange, was very anxious to acquaint him with the system which he had installed and which had proven so successful. ...Stevin's book becomes very important to Americans, because he materially influenced the views of his friend Richard Dafforne, who through his book "The Merchants' Mirrour," published in 1636, became practically the English guide and pioneer writer of texts on bookkeeping."

"Multiplication of whole Numbers ...Note, that for the more easie solution of this proposition, it were necessary to have in memory the multiplication of the 9 simple Characters among themselves, learning them by rote out of the Table here placed..."

"The seventh Definition. The Golden Rule, or Rule of three, is that by which to three tearmes given, the fourth proportionall tearme is found."

"The chapter of this valuable book... (...which I had reprinted... as an Appendix to Mr. Frend's Principles of Algebra,) relates to the method invented by Simon Stevinus... for finding... the first near value of x, or the root of any proposed Algebräick equation, by repeated conjectures and trials with easy numbers of one or two decimal figures: after having found which we may proceed to determine the value of the said root to a greater degree of exactness by one or more applications of Mr. Raphson's method of approximation. The title of this 10th Chapter of Mr. Kersey's Algebra is as follows: An Explanation of Simon Stevin's General Rule to extract one Root out of any possible Equation of Numbers, either exactly or very nearly true."

"It was not until about 1600 that the idea of writing fractions in the form of decimals was promoted in Western Europe. The Dutch mathematician Simon Stevin was the first to throw clear light upon the advantages of the decimal notation. In a pamphlet, De Thiende (The Dime), he advocated the use of decimal fractions. He urged that governments adopt the decimal system and also decimal coins, weights, and measures. This did not happen on a large scale, however, until the French Revolution."

"We find also the Famous ', Mathematician to the Prince of Orange, having defined Number to be, That by which is explained the quantity of every Thing, he becomes so highly inflamed against those that will not have the Unit to be a Number, as to exclaim against Rhetoric, as if he were upon some solid Argument. True it is that he intermixes in his Discourses a question of some Importance, that is, whether a Unit be to Number, as a Point is to a Line. But here he should have made a distinction, to avoid the confusing together of two different things. To which end these two questions were to have been treated apart; whether a Unit be Number, and whether a Unit be to Number, as a Point is to a Line; and then to the first he should have said, that it was only a Dispute about a Word, and that an Unit was, or was not a Number, according to the Definition, which a Man would give to Number. That according to Euclid's Definition of Number; Number is a Multitude of Units assembled together: it was visible, that a Unit was no Number. But in regard this Definition of Euclid was arbitrary, and that it was lawful to give another Definition of Number, Number might be defined as Stevin defines it, according to which Definition a Unit is a Number; so that by what has been said, the first question is resolved, and there is nothing farther to be alleged against those that denied the Unit to be a Number, without a manifest begging of the question, as we may see by examining the pretended Demonstrations of Stevin. The first is, The Part is of the same Nature with the whole, The Unit is a Part of a Multitude of Units, Therefore the Unit is of the same Nature with a MuItitude of Units, and consequently of Number. This Argument is of no validity. For though the part were always of the same nature with the whole, it does not follow that it ought to have always the same name with the whole; nay it often... has not the same Name. A Soldier is part of an Army, and yet is no Army... a Half-Circle is no Circle... if we would we could not... give to Unit more than its name of Unit or part of Number. The Second Argument which Stevin produces is of no more force. If then the Unit were not a Number, Subtracting one out of three, the Number given would remain, which is absurd. But... to make it another Number than what was given, there needs no more than to subtract a Number from it, or a part of a Number, which is the Unit. Besides, if this Argument were good, we might prove in the same manner, that by taking a half Circle from a Circle given, the Circle given would remain, because no Circle is taken away. ... But the second Question, Whether an Unit be to Number, as a Point is to a Line, is a dispute concerning the thing? For it is absolutely false, that an Unit is to number as a point is to a Line. Since an Unit added to number makes it bigger, but a Line is not made bigger by the addition of a point. The Unit is a part of Number, but a Point is no part of a Line. An Unit being subtracted from a Number, the Number given does not remain; but a point being taken from a Line, the Line given remains. Thus doth Stevin frequently wrangle about the Definition of words, as when he perplexes himself to prove that Number is not a quantity discreet, that Proportion of Number is always Arithmetical, and not Geometrical, that the Root of what Number soever, is a Number, which shews us that he did not properly understand the definition of words, and that he mistook the definition of words, which were disputable, for the definition of things that were beyond all Controversy."

"Diophantus is modern."

"The second Definition. Number is that which expresseth the quantitie of each thing."

"Deligne’s method was totally perpendicular to Grothendieck’s: he knew every trick of his master’s trade by heart, every concept, every variant. His proof, given in 1974, is a frontal attack and a marvel of precision, in which the steps follow each other in an absolutely natural order, without surprises. Those who heard his lectures had the impression, day after day, that nothing new was happening–whereas every lecture by Grothendieck introduced a whole new world of concepts, each more general than the one before–but on the last day, everything was in place and victory was assured. Deligne knocked down the obstacles one after the other, but each one of them was familiar in style. I think that this opposition of methods, or rather of temperament, is the true reason behind the personal conflict which developed between the two of them. I also think that the fact that “John, the disciple that Jesus loved” wrote the last Gospel by himself partly explains Grothendieck’s furious exile that Grothendieck has imposed upon himself."

"In Récoltes et Semailles, Grothendieck counts his twelve disciples. The central character is Pierre Deligne, who combines in this tale the features of John, “the disciple whom Jesus loved”, and Judas the betrayer. The weight of symbols!"

"Usually mathematicians are either theory builders, who develop tools, or problem-solvers, who use those tools to find solutions... Deligne is unusual in being both. He’s got a very special mind."

"The nice thing about mathematics is doing mathematics."

"From Grothendieck] and his example, I have also learned not to take glory in the difficulty of a proof: difficulty means we have not understood. The ideal is to be able to paint a landscape in which the proof is obvious."

"Over and above the specific theorems created by men such as Desargues, Pascal and La Hire, several new ideas and outlooks were beginning to appear. The first is the idea of continuous change of a mathematical entity from one state to another... [i.e., of a] a geometrical figure. It was Kepler, in his Astronomiae Optica of 1604, who first seemed to grasp the fact that parabola, ellipse, hyperbola, circle, and the degenerate conic consisting of a pair of lines are continuously derivable from each other. ...The notion of a continuous change in a figure was also employed by Pascal. He allowed two consecutive vertices of his hexagon to approach each other so that the figure became a pentagon. In the same manner he passed from pentagons to quadrilaterals. The second idea to emerge from the work of the projective geometers is that of transformation and invariance. To project a figure from some point and then take a section of that projection is to transform the figure to a new one. The properties... of interest are those that remain invariant under transformation. Other geometers of the seventeenth century, for example, Gregory of St. Vincent... and Newton, introduced transformations other than projection and section."

"Gregory St. Vincent is the greatest of circle-squarers, and his investigations led him into many truths: he found the property of the arc of the hyperbola which led to Napier's logarithms being called hyperbolic. Montucla says of him, with sly truth, that no one ever squared the circle with so much genius, or, excepting his principal object, with so much success."

"Grégoire... was primarily concerned to illustrate by reference to the ungula that volumetric integration could be reduced, through the ductus in planum, to a consideration of geometric relations between the lines of plane figures. ...Unfortunately, the delayed publication of the Opus geometricum prevented it from receiving... attention... In 1647, ten years after the publication of Descartes' La Géométrie, algebraic methods were rapidly gaining ground and the form and manner of presentation of Grégoire's work was not such as to make easy reading. ...Amongst those who gained much from the Opus geometricum... [was] Blaise Pascal whose Traité des trilignes rectangles et de leurs onglets is based essentially on the ungula of Grégoire. Huygens recommended the section on geometric series to Leibniz who later came to make a thorough study of the entire work. Tschirnhaus... found in the ductus in planum a valuable foundation for the development of his own algebraic integration methods."

"No one ever squared the circle with so much ability or (except for his principal object) with so much success."

"Grégoire de Saint-Vincent, a Jesuit, born in Bruges in 1584 and died in Ghent in 1667, discovered the expansion of log(1+x) in ascending powers of x. Although a circle squarer, he is worthy of mention for the numerous theorems of interest which he discovered in his search after the impossible... He wrote two books on the subject [1647, 1668]... the fallacy in the quadrature was pointed out by Huygens. In the former work he used indivisibles. An earlier work entitled Theoremata Mathematica, published in 1624, contains a clear account of the method of exhaustions, which is applied to several quadratures, notably that of the hyperbola."