Gauss, Carl Friedrich

A BIBLIOGRAPHY OF GAUSS

SUPPLEMENTARY BIBLIOGRAPHY

Carl Friedrich Gauss (1777–1855), greatest of German mathematicians, was born in Brunswick on April 30, 1777. (He was baptized Johann Friedrich Carl, but he later dropped his first name and reversed the second and third.) Ranked with Archimedes and Newton as one of the three greatest mathematicians of all time, he combined in a most unusual way a pure mathema tician’s interest in abstract ideas and logical rigor, a theoretical physicist’s interest in the creation of mathematical models of the physical world, an astronomer’s talent for keen observation, and an experimentalist’s skill in the application and invention of methods of measurement. He was blessed with a stupendous faculty for mental calculation, which enabled him to explore numerical relationships experimentally and to carry out extensive or involved routine computations quickly and accurately; he also had a gift for learning ancient and modern languages, which became his hobby. He contributed mightily to every branch of pure and applied mathematics that existed in his day, some of which he had founded, and he made major contributions in astronomy, geodesy, physics, and metrology. Each of his many interests had its principal, but not exclusive, season: before 1800, philology and number theory; 1800-1820, astron omy; 1820-1830, geodesy, differential geometry, and conformal mapping; 1830-1840, geomagnetism, electromagnetism, and general theory of in verse-square forces; 1840-1855, topology and the geometry of functions of a complex variable. He died in Gottingen on February 23, 1855. Of his many contributions, those used most widely in the physical, biological, and social sciences today relate to the method of least squares, his first formulation of which dates from 1795 to 1798 and his second from 1821 to 1823.

Gauss was the only child of Gebhard Dietrich Gauss, a bricklayer and gardener, by his second wife, nee Dorothea Benze, daughter of a stone mason. Gebhard Gauss, being skilled in writing and calculating, kept accounts for a local insurance company; he was esteemed by the townspeople, but at home he was harsh and uncouth, which repelled his brilliant son. Gauss’s mother had no special schooling, could not write, and could barely read, but she was cheerful, intelligent, and of strong character. She had a genial and extraordinarily intelligent younger brother, Johann Friedrich, a skilled weaver of artistic damasks, who quickly spotted his nephew’s unusual talents and capacities and enjoyed sharpening his wits on those of his sister’s young genius. Gauss, as a small boy, thought highly of his uncle and later, lamenting his untimely death in 1809, often declared that a born genius had been lost in him.

Gauss’s precocity is unequaled in the history of mathematics. Before he was three, while watching his father’s payroll calculations he detected an error and announced the correct result. He was admitted at age ten to the beginners’ class in arithmetic at St. Katherine’s Volksschule in Brunswick, where the speed and accuracy of his mental calculations so astonished the schoolmaster that he purchased for the boy the best obtainable textbook on arithmetic, which Gauss quickly mastered, convincing the schoolmaster that Gauss had gone beyond him. Luckily the schoolmaster had an assistant who was interested in mathematics. He and Gauss studied algebra and the rudiments of calculus together in the evenings, helping each other over difficulties and amplifying the textbook ?proofs.? Thus in his eleventh year Gauss became acquainted with the binomial theorem and, finding the textbook “proof” unsatisfactory when the exponent n is not a positive integer, devised his own proof of the convergence of the infinite series involved, which established him as one of the first of the “rigorists” and served as an inspiration for some of his greatest later work.

Early in 1791 Gauss’s amazing powers came to the attention of Carl Wilhelm Ferdinand, duke of Brunswick, who became Gauss’s patron, paid the expenses of his education at Caroline College from 1792 to 1795 and at the University of Göttingen from 1795 to 1798, and until his death in 1806 gave Gauss considerable additional financial support. At Caroline College, Gauss devoted himself with equal success to classical literature, philosophy, and advanced mathematics. He studied carefully the original works of Newton, Euler, and Lagrange. In March 1795 (Dunnington 1955, p. 391) he rediscovered that invaluable principle of number theory, the law of quadratic reciprocity, which Legendre had published in 1785, and of which Gauss was to publish the first rigorous proof in 1801. Yet, on entering the University of Gottingen in October 1795, he was still undecided whether to make mathematics or philology his career.

Mathematics won on March 30, 1796, the day Gauss discovered that a regular polygon of 17 sides is amenable to straightedge-and-compass construction; such a construction had eluded mathematicians for two thousand years. By June 1, 1796, he had discovered much more: a regular polygon with an odd number of sides is amenable to such construction if and only if the number of its sides is a prime Fermat number, F_n = 2²ⁿ + 1, or a product of different prime Fermat numbers (five of which are known today). In 1796 he also developed the first rigorous proof of the fundamental theorem of algebra (i.e., that every nonconstant polynomial has a root), which became the subject of the doc toral dissertation for which he was awarded a PH.D. in absentia by the University of Helmstedt in 1799. In the autumn of 1798 Gauss polished the final draft of his greatest masterpiece, Disquisitiones arithmeticae, the printing of which began in 1799 but was not completed until September 1801, owing to the sale of the original print shop. Gauss brought together in this work his own original contributions to the theory of integral numbers and rational fractions and all of the principal related, but somewhat disconnected, results of his predecessors, so enriching the latter by rigorous reformulation and blending into a unified whole that this great book is regarded as marking the beginning of the theory of numbers as a separate, systematic branch of mathematics.

Among Gauss’s early achievements was his reduction of the ecclesiastical calendar’s extremely complicated computational procedure for finding the date of Easter (see Encyclopaedia Britannica, 11th ed., vol. 4, pp. 991-999) to a set of simple formulas from which the answer for any given year can be found in a few minutes. His motivation stemmed from his mother’s inability to recall the exact date of his birth—only that it was a Wednesday, eight days before Ascension Day. As first published (1800), his procedure gave an incorrect date for Easter in 1734 and would have been incorrect again in 1886. Gauss supplied the necessary correction in 1807 and later provided an additional correction needed from 4200 on. (A semipopular exposition of Gauss’s procedure, with a worked example, has been provided by H. Herbert Howe [1954].)

Gauss’s first formulation of the method of least squares dates from his student days in the autumn of 1794 he read (Galle 1924, p.5)Lambert’s discussion (1765, pp.428-488)of the determination of the coefficients of linear relationship y = ?+ ?_x from a set of n (>2) observational points Y_i,x_i)by the method of averages. In 795 Gauss conceived the simpler and more objective procedure of taking for ? nd ? he values a and b the values a and b that minimize the sum of squared residuals, ?_i(Y_i — a — bx_i)², and worked out the computational details, except for the weighting of observations of unequal precision (Galle 1924, p. 7). In 1797 he attempted to justify his minimum-sum-of-squared-residuals (mssr) technique by means of the theory of probability but “found out soon that determination of the most probable value of an unknown quantity is impossible unless the probability distribution of errors of observation is known explicitly” ([1821a] 1880, p. 98). Therefore, “it seemed to him most natural to proceed the other way around” and to seek the probability distribution of errors that “in the simplest case would result in the rule, generally accepted as good, that the arithmetic mean of several values for the same unknown quantity obtained from equally reliable ob servations shall be considered the most probable value” ([1821a] 1880, p. 98).

The concept of a probability distribution, or “law,” of errors originated with Thomas Simpson (1755): he studied the sampling distribution of the arithmetic mean of n independent and identically distributed errors subject to a discrete rectangular or a discrete triangular law of error, and concluded that “the more observations or experi ments are made, the less will the conclusion be subject to err, provided they admit of being made under the same circumstances” (p. 93). In 1757 he extended his analysis to samples of n from a continuous triangular distribution. Studies of other continuous laws of error followed: rectangular, by Lagrange, published in 1774; double-exponential (Ce-^m|x| -?<x<+?), by Laplace in 1774; semicircular , by Daniel Bernoulli in 1778; and double-logarithmic (C log a/|x|,— a ?x ?+ a), by Laplace in 1781 (for further details, see Eisenhart 1964; Todhunter 1865; Merriman 1877).

Using inverse probability, Gauss found f(v) = (h/??) exp h²v²) to be the required probability distribution of errors (1809, art. 175-177). He not ed that h (— 1/(??2) in modern notation, where cr is the root-mean-square error, or standard deviation, of the distribution) “can be considered as the measure of precision of the observations” (art. 178). Although De Moivre in 1733 had adduced the function e^t2 to approximate sums of successive terms of the binomial expansion of (a + b)ⁿ when is large (for what he actually wrote, see Smith [1929] 1959, pp. 566-575), and Laplace, in a series of papers published during the period 1774-1786, had explored in great detail the use of this function and its derivatives to approximate various probability distributions arising in games of chance, notably the binomial and hypergeometric distributions and the corresponding incomplete beta-function forms obtained through application of Bayes’ theorem, and had suggested tabulation of its integral for use in such problems (see Todhunter [1865] 1949, art. 890-911) neither De Moivre nor Laplace seems to have considered C exp (— h²x²), or an equivalent expression, as a law of error or as a probability distribution in its own right.

Gauss then went on to deduce his mssr technique from f(v), by what we would today call the method of maximum likelihood, finding that when the respective observations, Y,i = 1, 2,^…, n, are of unequal precision, each residual should be multiplied by the corresponding measure of precision, hi, i = 1, 2,^…, n, before squaring (art. 179); derived the formula for the precision, h(Y?), of any weighted arithmetic mean, Y?, of n independent observations (art. 181), finding that the precision, h(Y?), of the unweighted arithmetic mean, Y?, of n equally precise independent observations is proportional to ?n(art. 173); derived for the case of observations on linear functions of several unknown quantities ?, ?,^… the rule of formation of (what we call) the normal equations that jointly determine optimal estimators A, B, ^… (art. 180); outlined (art. 182) his famous method of elimination for solving the normal equations—most widely known today through a modification published by M. H. Doolittle (1878)—that provides as a by product an easily evaluated expression (art. 182, sec. 5) for the corresponding minimum sum of squared residuals in terms of quantities found in the course of their solution; and gave similar expressions (art. 183) for evaluating the precisions h_A, h_B, ^… of A, B, ^… as determinations of ?, ?, ^….

An entry in Gauss’s diary (see Werke, vol. 10, p. 533) indicates that he completed the foregoing development of the method of least squares on June 17, 1798. A fragment of his first application of these procedures in the spring of 1799 has been found (see Werke, vol. 12, pp. 64-68), and a short note ([1799] 1900, p. 136) dated August 24, 1799, on another application was published in October 1799.

In 1801 Gauss became deeply involved in the development and application of new astronomical methods which, in conjunction with his least squares techniques, resulted in amazingly accurate predictions, despite the scanty data available, of the orbits of some newly discovered planets; the success of his new methods brought about Gauss’s immediate recognition as a first-rank astronomer (Bell 1937; Dunnington 1955, pp. 49-57). In 1805 Gauss began preparation (in German) of his second masterpiece, Theoria motus corporum coelestium … (1809), in which he was to give a complete system of formulas and procedures for computing the motion of a body whose orbit is a conic section, and a general method for determining the orbit of a comet or a planet from only three observed positions. The third section (art. 172-189) he devoted to a detailed exposition of the theory and application of his least squares methods. The German text was completed in 1806 (Galle 1924, p. 10). When the work was offered for publication in 1807, the publisher accepted it only on condition that Gauss translate it into Latin, because of the very unsettled political situation in Germany following the disastrous defeat of the Prussian army (under the leadership of Gauss’s patron, the duke of Brunswick) by the Napoleonic forces at the battle of Auerstedt in October 1806. Consequently, full details of Gauss’s first formulation of the method of least squares were not published until the Latin translation appeared in 1809. In the meantime, Legendre’s independent formulation of his methods of least squares had appeared in print (1805); this went little beyond what Gauss had developed in 1795—no probability considerations are involved, and there is no discussion of precision or weighting of observations. A controversy followed, bitter on Legendre’s part, in which many persons became involved, and which, as Bell explains, “was most unfortunate for the future development of mathematics” ([1937] 1956, p. 331).

In his 1809 presentation, Gauss regarded the measures of precision h₁, h₂,, … of the respective observations involved as known quantities and said nothing about how to determine their values in practice. F. W. Bessel took the first step in this direction: in 1815 he introduced the probable error as a measure of imprecision (1815, p. 234), which he later defined (1816, p. 142) as the magnitude, r, that an error has an equal chance of exceeding (or being less than) in absolute value; and from 48 determinations of the right ascension of the polestar he obtained a value for the probable error of such determinations (1815, p. 234) by means of the formula R = 0.8453?_i|Y_i-Y?|/n, where Y? = ?_iY_i/n. (The numerical result given conforms to this formula [cf. Gauss 1816, art. 8], which is not stated.) The following year, Bessel (1816, pp. 141-142) showed that the probable error for Gauss’s error distribution, f(v), is given by r = 0.47693h-¹ = 0.8453? = 0.6745?. In this formulation ?= l/(h??) represents the mean absolute error of the distribution and ?= 1/(h?2) is the root-mean-square error of the distribution, respectively. Gauss immediately pointed out (1816, art. 3) that given m independent errors V₁, V₂,^…, V_m distributed according to his error function, f(v), then for “m large or small” the “most probable values” (or, in modern terminology, the maximum likelihood estimators) of h and r are and R? = 0.6744897, respectively. He showed (art. 4) that for large m, ? — h and — r are distributed approximately as f(v), with h_H ?m/h and h_R = ?m/r, and gave (loc. cit.) explicit expressions, in terms of ? and R?, for the “probable limits” of (in modern terminology, 50 per cent confidence limits for) “the true values of h and r.” Next he considered (art. 5) the estimation of r by with C_p =C_p(m,h) chosen so that r is the mean of the large-sample distribution of R_p, and found R₂ to be the most precise, noting (art. 6) that R₂ for m = 100 is as precise a measure of r as is R₁, for m = 114, R, for m = 109, R₄ for m = 133, R₅ for m = 178, and R₆ for m = 251. Finally, he considered (art. 7) the estimation of r by the “middlemost” (i.e., the median) of the absolute errors |V_i| when m is odd and found that this procedure requires m = 249 in order to achieve precision comparable to R₂ for m = 100. Gauss then evaluated (art. 8) R_1; R₂, R₃, and R₄ for the 48 determinations considered by Bessel (1815), taking the residuals, Y_i - Y?, as measures of the corresponding errors, V_i, and concluded that Bessel had used the formula R₁.

On February 15, 1821, Gauss presented to the Royal Academy of Sciences in Göttingen a completely new formulation of the method of least squares that was entirely free from dependence on any particular probability distribution of errors. Entitled “Theoria combinationis observationum erroribus minimis obnoxiae. Pars prior” (1821a), it stemmed from the earlier work of Laplace: Three years before Gauss’s birth, Laplace ([1774] 1891, pp. 41-48) had said that to estimate a parameter ?one ought to use that function T - T(Y₁,Y₂, …) of the observations, Y₁, Y₂, …, for which the mean (or expected) absolute error of estimation, E{|T — ?|}, is a minimum for the given probability distribution of errors, and, for the case of independent identically distributed observations, Laplace gave an explicit procedure for finding such a function for estimating the location parameter of their common distribution, f(y - ?), when this is completely specified except for the value of T. Finally, Laplace showed that in the case of independent observations of equal (or unequal) precision, Gauss’s technique of minimizing the sum of squared residuals leads to the same estimators as Laplace’s own procedure for minimizing the mean absolute error of estimation when and only when (Laplace [1812] 1820, book 2, chapter 4, sec. 23) the errors X_t = Y_i — ? of the respective observations are distributed in accordance with Gauss’s law of error,

Gauss ([1821a] 1880, art. 6-7) proposed using in stead the mean square error of an observation, E{(Y-?)²}, or of an estimator E{(T-?)²}, as a better measure of “uncertainty” (”incerti-tudo“) and derived (art. 9) his remarkable inequalities (see Savage 1961, eq. 9) for the probability that a random variable Z with a continuous unimodal probability distribution will differ (positively or negatively) from its modal value z₀ by more than ? times its root-mean-square error measured from z_n. Then he showed (art. 18-23) that when the mean values E{Y_i} — ?_i of a set of n independ ent observations Y_lY₂, …, Y_n are linear functions of k (< n) unknown parameters ?_j (j = 1, 2,…, k) and (in modern terminology) the variances E(Y_i - ?_i)²} = ?_i² of the Y; are all finite, then the mssr technique yields estimators T, of the ?, having minimum mean-square error (j = 1, 2, …, k), whatever the distribution(s) of errors of the ob servations. In a second memoir (1823), he extended the preceding result to estimators L_r of linear functions A_r(?₁, ?₂,…, ?_k) of the ?’s (r=l,2,…, m; m?k) and showed (art. 37-40) that the resultant minimum sum of squared residuals is strictly equivalent to the sum of n - m independent errors (i.e., has n - m degrees of freedom). These results, at one time attributed erroneously to A. A. Markov (for explanation, see Neyman ([1938] 1952, p. 228), are derived and discussed by various recent authors (e.g., Graybill 1961; Scheffé 1959; Zelen 1962) as the “Gauss-Markov theorem,” and the method of least squares is thereby endorsed as a procedure for obtaining minimum variance linear unbiased estimators.

Gauss definitely preferred his 1821 formulation of the method of least squares above all others (for his statement to this effect on February 26, 1821, see [1821b] 1880, p. 99). In a letter to F. W. Bessel (Gauss & Bessel 1880) dated February 28, 1839 (an excerpt from which is given in his Werke, vol. 8, pp. 146-147), he remarked that he had never made a public statement of his reasons for abandoning the metaphysical approach of his first formulation but that a decisive reason was his belief that maximizing the probability of a zero error is less important than minimizing the probability of committing large errors.

In his later years, Gauss took special pride in his contributions to the development of the method of least squares and, despite a lifelong aversion to teaching, gave a course on this subject at the University of Gottingen each year from 1835 until his death. During his lifetime, the method of least squares became a basic tool in astronomy and geodesy throughout the world and has remained one to this day. And when Karl Pearson and G. Udny Yule began to develop the mathematical theory of correlation in the 1890s, they found that much of the mathematical machinery that Gauss devised for finding “best values” for the parameters of empirical formulas by the method of least squares was immediately applicable in correlation analysis, in spite of the fact that the aims of correlation analysis are the very antithesis of those of the theory of errors. As Galton remarked in his Memories of My Life (1908, p. 305), “The primary objects of the Gaussian Law of Error were exactly opposed, in one sense, to those to which I applied them. They were to get rid of, or to provide a just allowance for errors. But these errors or deviations were the very things I wanted to preserve and to know about.” In consequence, Gauss’s contributions to the method of least squares embody mathematics essential to statistical theory and its applications in almost every field of science today.

Gauss was visited in 1829 by the Belgian astronomer-physicist-statistician Adolphe Quetelet, who was engaged in making geomagnetic measurements in Holland, Germany, Italy, and Switzerland. Gauss had been interested in geomagnetism from around the turn of the century and had remarked, in a letter to Heinrich Olbers dated March 1, 1803, “I believe that this offers a greater field for the application of mathematics than has yet been supposed” (Olbers & Gauss, Briefwechsel. …); his intense activity in other fields had thus far prevented his undertaking research in geomagnetism, despite the strenuous efforts of Alexander von Humboldt, in 1804 and again in 1828, to persuade him to do so. When Quetelet arrived, he found Gauss studying Russian for relaxation: “I have been very fatigued,” he said, “from occupying myself with astronomy, geodesy, and other subjects that I know fairly well; I wanted to turn my attention to a language that I did not know at all, and now I am reading Russian” (Quetelet 1866, p. 646). Measurement of the intensity of the earth’s magnetism was new to Gauss, and he was eager to know how such measurements were made and the precision that could be achieved (p. 645). Quetelet set up his apparatus in Gauss’s yard, and together they conducted a series of experiments, taking observations simultaneously but in slightly different manners. The agreement of the results obtained astonished Gauss, who exclaimed: “But these observations conform to the precision of astronomical observations” (p. 646).

From January 1831 on, geomagnetic measurements were made regularly at Gottingen. By Febru ary 1832, Gauss was deeply involved in research on geomagnetism and had found that he could express the intensity of geomagnetism in what he called “absolute units,” that is, in terms of units of the three fundamental physical quantities: length, mass, and time. On December 15, 1832, he presented his findings to the Royal Academy of Sci ences in Gottingen in a paper entitled “Intensitas vis magneticae terrestris ad mensuram absolutam revocata” (1832), which was promptly recognized as one of the most important papers of the century.

Gauss made electromagnetic measurements for the first time in October 1832. By Easter 1833, he and his young physicist colleague Wilhelm Weber had put into operation an electromagnetic telegraph between Gauss’s observatory and Weber’s physics laboratory (Dunnington 1955, pp. 147-148, 395). They sent only individual words at first, and later complete sentences. Plans were drawn up in 1835-1836 for its use on the Leipzig-Dresden railroad but were dropped when the railroad authorities declared that the wires would have to be put underground. A monument showing Gauss and Weber discussing their telegraph was erected on the campus of the University of Göttingen.

In 1832 Gauss had begun preparation of his “Allgemeine Theorie des Erdmagnetismus” (”general Theory of Terrestrial Magnetism“); completion was delayed by lack of experimental mate rial. At Gauss’s suggestion a magnetic observatory (of nonmagnetic construction) was erected at the University of Göttingen in 1833. By 1836 Göttingen had become the principal European center for research on geomagnetism; and an association of magnetic observatories, known as the Göttingen Magnetic Union, was formed to coordinate simultaneous measurement of geomagnetic phenomena throughout Europe. In 1837 Gauss and Weber collaborated in the invention of a galvanometerlike device, the bifilar magnetometer, for measuring magnetic field intensities; and with it, Gauss verified the inverse-square law of magnetic attraction to which he had already been led by theory. His “Allgemeine Theorie . . .” appeared in 1839 and was followed in 1840 by his (with Weber) “Atlas des Erdmagnetismus” (1840) and his great treatise “Allgemeine Lehrsätze in Beziehung auf die im verkehrten Verhaltnisse . . .” (1840)—i.e., what today we call “potential theory“—which marked the peak of his work in physics and the close of his work on magnetism. Near the end of his life, Gauss, like Laplace, Fourier, and Poisson, turned his attention to the social sciences and to the help that they might derive from the physical sciences. In particular, he took an active interest in application of the theory of probability to social laws. Thus, in 1847, he corresponded with the Danish astronomer Heinrich Christian Schumacher on the laws of mortality and on the construction of mortality tables (Quetelet 1866, pp. 653-655).

Called “the prince of mathematicians” even in his lifetime, Gauss received a steady stream of honors (listed chronologically in Dunnington 1955, appendix B), beginning with his election in 1802 as a corresponding member of the Imperial Academy of Arts and Sciences in St. Petersburg. He was elected a fellow of the Royal Society of London in 1804 and received the Copley Medal in 1838; he became a full member of the Royal Academy of Sciences in Berlin in 1810, a fellow of the American Academy of Arts and Sciences in Boston, Mas sachusetts, in 1822, and a member of the American Philosophical Society in Philadelphia in 1853. In 1842 the highest order conferred by the kingdom of Prussia, Pour le mérite, was awarded to him. On July 16, 1849, exactly 50 years after receipt of his doctorate, a Gauss jubilee was held in Got tingen at which honors were showered upon him, including honorary citizenship of Brunswick and Göttingen, which he prized above all the rest.

Churchill Eisenhart

[For the historical context of Gauss’s work, see the biographies ofLaplace; Moivre; for discussion of the subsequent development of his ideas, seeEstimation; Linear Hypotheses, article onRegression; MultivariateAnalysis, articles onCorrelation; and the biographies ofPearson; Quetelet; Yule.]

A BIBLIOGRAPHY OF GAUSS

WORKS BY GAUSS

(1799) 1900 Zur Geschichte der Entdeckung der Methode der kleinsten Quadrate. Volume 8, pages 136-141 in Carl Friedrich Gauss Werke. Göttingen: Dieterichsche Universitäts-Druckerei.

(1800) 1874 Berechnung des Osterfestes. Volume 6, pages 73-79 in Carl Friedrich Gauss Werke. Göttingen: Dieterichsche universitäts-Druckerei.

(1801) 1966 Disquisitiones arithmeticae. English translation by Arthur A. Clarke. New Haven: Yale Univ. Press.

(1807) 1874 Noch etwas über die Bestimmung des Oster festes. Volume 6, pages 82-86 in Carl Friedrich Gauss Werke. Göttingen: Dieterichsche universitäts-Druck erei.

(1809) 1963 Theory of Motion of the Heavenly Bodies Moving About the Sun in Conic Sections. New York: Dover. ? First published as Theoria motus corporum coelestium….

(1816) 1880 Bestimmung der Genauigkeit der Beobachtungen. Volume 4, pages 109-117 in Carl Friedrich Gauss Werke. Göttingen: Dieterichsche universitäts-Druckerei. ? An English translation appears in Gauss’s Work (1803–1826). See also section 3 of Whittaker and Robinson 1924.

(1821a) 1880 Theoria combinationis observationum erroribus minlmis obnoxiae. Pars prior. Volume 4, pages 1-26 in Carl Friedrich Gauss Werke. Göttingen: Diet erichsche universitäts-Druckerei. ? An English translation appears in Gauss’s Work (1803–1826).

(1821b) 1880 Anzeigen: Theoria combinationis observationum erroribus minimis obnoxiae. Pars prior. Volume 4, pages 95-100 in Carl Friedrich Gauss Werke. Göttingen: Dieterichsche universitäts-Druckerei.

(1823) 1880 Theoria combinationis observationum erroribus minimis obnoxiae. Pars posterior. Volume 4, pages 27-53 in Carl Friedrich Gauss Werke. Gbt tingen: Dieterichsche universitäts-Druckerei. ? An English translation is in Gauss’s Work (1803–1826).

(1825–1827) 1965 General Investigations of Curved Surfaces. Hewlett, N.Y.: Raven Press. ? Translations of Gauss’s 1827 paper “Disquisitiones generales circa superficies curvas,” his abstract of it, and his 1825 fragment “Neue allgemeine Untersuchungen uber die krummen Flachen.”

(1832) 1877 Intensitas vis magneticae terrestris ad mensuram absolutam revocata. Volume 5, pages 79-118 in Carl Friedrich Gauss Werke. Göttingen: Dieterichsche universitäts-Druckerei. ? Magie 1935 contains English excerpts.

(1839) 1966 General Theory of Terrestrial Magnetism. Volume 2, pages 184-251 in Richard Taylor (editor), Scientific Memoirs, Selected From the Transactions of Foreign Academies of Science, and Learned Societies, and From Foreign Journals. New York: Johnson. ? First published in German.

(1840) 1966 General Propositions Relating to Attractive and Repulsive Forces Acting in the Inverse Ratio of the Square of the Distance. Volume 3, pages 153-196 in Richard Taylor (editor), Scientific Memoirs, Selected From the Transactions of Foreign Academies of Science, and Learned Societies, and From Foreign Journals. New York: Johnson. ? First published in German.

(1840) 1929 Gauss, Carl Friedrich; and Weber, Wil-Helm Atlas des Erdmagnetismus nach den Elementen der Theorie entworfen: Supplement zu den Resultaten aus den Beobachtungen des magnetischen Vereins unter Mitwirkung von C. W. B. Goldschmidt. Volume 12, pages 335-408 in Carl Friedrich Gauss Werke. Göttingen: Dieterichsche universitäts-Druckerei. ? Charts following page 408.

COLLECTIONS OF GAUSS’S WORKS

1855 Méthode des moindres carrés: Mémoires sur la combinaison des observations. Translated by J. Bertrand, and published with the authorization of the author. Paris: Mallet-Bachelier.

Abhandlungen zur Méthode; der kleinsten Quadrate. The saurus mathematicae, Vol. 5. Würzburg: Physica-Verlag, 1964.

Carl Friedrich Gauss Werke. 12 vols. Göttingen: Dieter ichsche universitäts-Druckerei, 1870–1933.

Gauss’s Work (1803–1826) on the Theory of Least Squares. Translated by Hale F. Trotter. Statistical Techniques Research Group, Technical Report, No. 5. Princeton, N.J. Princeton Univ., 1957. ? Prepared from Gauss 1855.

WORKS CONTAINING EXTRACTS BY GAUSS

Magie, William F. (1935) 1963 A Source Book in Physics. Cambridge, Mass.: Harvard Univ. Press. ? See especially extracts from Gauss 1832.

Midonick, Henrietta O. (editor) 1965 The Treasury of Mathematics: A Collection of Source Material in Mathematics. New York: Philosophical Library.

Smith, David Eugene (1929) 1959 A Source Book in Mathematics. 2 vols. New York: Dover.

Taylor, Richard (editor) 1966 Scientific Memoirs, Selected From the Transactions of Foreign Academies of Science, and Learned Societies, and From Foreign Journals. 7 vols. New York: Johnson.

CORRESPONDENCE

Briefe von C. F. Gauss an B. Nicolai. Karlsruhe: Braun,1877.

Gauss, Carl Friedrich; and Bessel, Friedrich W. Brief wechsel zwischen Gauss und Bessel. Leipzig: Engelmann, 1880.

Gauss, Carl Friedrich; and Bolyai, WolfgangBrief wechsel zwischen Carl Friedrich Gauss und Wolfgang Bolyai. Leipzig: Teubner, 1899.

Gauss, Carl Friedrich; and Gerling, Christian L. Brief wechsel zwischen Carl Friedrich Gauss und Christian Ludwig Gerling. Berlin: Eisner, 1927.

Gauss, Carl Friedrich; and Schumacher, Heinrich C. Briefwechsel zwischen C. F. Gauss und H. C. Schu macher. 6 vols. Altona: Esch, 1860–1865.

Humboldt, Alexander Von; and Gauss, Carl F. Briefe zwischen A. v. Humboldt und Gauss. Leipzig: Engelmann, 1877.

Olbers, Wilhelm; and Gauss, Carl F. Briefwechsel zwischen Olbers und Gauss. 2 parts. Berlin: Springer, 1900–1909. ? Published as Volume 2 of Wilhelm Olbers, Sein Leben und seine Werke.

SUPPLEMENTARY BIBLIOGRAPHY

Bell, Eric T. (1937) 1956 Gauss: The Prince of Math ematicians. Volume 1, pages 295-299 in James R. Newman (editor), The World of Mathematics: A Small Library of the Literature of Mathematics From A’h-mose the Scribe to Albert Einstein. New York: Simon & Schuster. ? A paperback edition was published in 1960.

Bessel, Friedrich W. 1815 Ueber den Ort des Polar-sterns. Astronomisches Jahrbuch [1818]:233—241.

Bessel, Friedrich W. 1816 Untersuchungen uber die Bahn des Olbersschen Kometen. Akademie der Wissenschaften, Berlin, Mathematische Klasse, Abhand-lungen [1812-1813]: 119–160.

Doolittle, M. H. 1878 [Method Employed in This Office in the Solution of Normal Equations and in the Adjustment of a Triangulation.] U.S. Coast and Geodetic Survey, Report of the Superintendent [1878], Appendix 8, Paper 3:115–120.

Dunnington, G. Waldo 1955 Carl Friedrich Gauss, Titan of Science: A Study of His Life and Work. New York: Hafner. ? Contains a comprehensive bibliography of works by and about Gauss.

Eisenhart, Churchill 1964 The Meaning of “Least” in Least Squares. Journal of the Washington Academy of Sciences 54:24–33.

Festschrift zur Feier der Enthiillung des Gauss—Weber-Denkmals in Gottingen. 2 vols. 1899 Leipzig: Teub-ner.

Galle, A. 1924 Über die geodätischen Arbeiten von Gauss. Volume 11, part 2, pages 1-165 in Carl Fried rich Gauss Werke. Göttingen: Dieterichsche Univer-sitäts-Druckerei.

Galton, Francis 1908 Memories of My Life. London: Methuen.

Graybill, Franklin A. 1961 An Introduction to Linear Statistical Models. Vol. 1. New York: McGraw-Hill.

HÃnselmann, Ludwig 1878 Karl Friedrich Gauss: Zwölf Kapitel aus seinem Leben. Leipzig: Duncker & Humblot.

Howe, H. Herbert 1954 How to Find the Date of Easter. Sky and Telescope 13:196.

Klein, Felix et al. 1911-1920 Materialien fur eine wis-senschaftliche Biographie von Gauss. 8 vols. Leipzig: Teubner.

Lambert, Johann Heinrich 1765 Beytrage zum Gebrauche der Mathematik und deren Anwendung. Vol. 1. Berlin: Buchladen der Realschule.

Laplace, Pierre Simon De (1774) 1891 Memoire sur la probability des causes par les evenements. Volume 8, pages 27-65 in Oeuvres complètes de Laplace. Paris: Gauthier-Villars. ? See especially “Probleme III: Determiner le milieu que Ton doit prendre entre trois observations donnees d’un meme phenomene” on pages 41–48.

Laplace, Pierre Simon De (1812) 1820 Theorie an-alytique des probabilites. 3d ed., rev. Paris: Cour-cier. ? Smith 1929 contains English extracts from Book 2.

Laplace, Pierre Simon DeOeuvres completes de Laplace. 14 vols. Paris: Gauthier-Villars, 1878–1912.

Legendre, Adrien M. (1805) 1959 On the Method of Least Squares. Volume 2, pages 576-579 in David Eugene Smith, A Source Book in Mathematics. New York: Dover. ? First published as “Sur la Méthode; des moindres quarrés” in Legendre’s Nouvelles me-thodes pour la détermination des orbites des cometes.

Mack, Heinrich (editor) 1927 Carl Friedrich Gauss und die Seinen: Festschrift zu seinem 150. Geburts-tage. Brunswick: Appelhans.

Merriman, Mansfield 1877 A List of Writings Relating to the Method of Least Squares, With Historical and Critical Notes. Connecticut Academy of Arts and Sciences, Transactions 4:151–232.

Moivre, Abraham De (1733) 1959 A Method of Ap proximating the Sum of the Terms of the Binomial Expanded Into a Series, From Whence Are Deduced Some Practical Rules to Estimate the Degree of Assent Which Is to Be Given to Experiments. Volume 2, pages 566-575 in David Eugene Smith, A Source Book in Mathematics. New York: Dover. ? First published as “Approximatio ad summam terminorum binomii in seriem expansi.”

Neyman, Jerzy (1938)1952 Lectures and Conferences on Mathematical Statistics and Probability. 2d ed. Washington: U.S. Department of Agriculture. ? See especially Chapter 4, “Statistical Estimation,” in the second edition.

Quetelet, Adolphe 1866 Charles-Frederic Gauss. Pages 643-655 in Adolphe Quetelet, Sciences mathematiques et physiques chez les Beiges au commencement du Xix” siecle. Brussels: Van Buggenhoudt.

Sartorius Von Waltershausen, Wolfgang 1856 Gauss zum Gedachtniss. Leipzig: Hirzel.

Savage, I. Richard 1961 Probability Inequalities of the Tchebycheff Type. U.S. National Bureau of Standards, Journal of Research, B. Mathematics and Mathematical Physics 65 B: 211–222.

Scheffe, Henry 1959 The Analysis of Variance. New York: Wiley.

Schering, Ernst (1877) 1909 Carl Friedrich Gauss’ Geburtstag nach hundertjahriger Wiederkehr. Pages 176-213 in Ernst Schering, Gesammelte mathemati sche Werke. Berlin: Mayer & Miiller.

Simpson, Thomas 1755 A Letter to the Right Honourable George Earl of Macelesfield, President of the Royal Society, on the Advantage of Taking the Mean of a Number of Observations, in Practical Astronomy. Royal Society of London, Philosophical Transactions 49, part 1:82–93.

Simpson, Thomas 1757 An Attempt to Show the Ad vantage Arising by Taking the Mean of a Number of Observations in Practical Astronomy. Pages 64-75 in Thomas Simpson, Miscellaneous Tracts on Some Curi ous and Very Interesting Subjects in Mechanics, Phys ical-astronomy, and Speculative Mathematics. London: Nourse.

Todhunter, Isaac (1865) 1949 A History of the Mathe matical Theory of Probability From the Time of Pascal to That of Laplace. New York: Chelsea.

Whittaker, E. T.; and Robinson, G. (1924) 1944 The Calculus of Observations: A Treatise on Numerical Mathematics. 4th ed. Princeton, N.J.: Van Nostrand. ? Section 103 provides a digest of Gauss 1816.

Winnecke, F. A. T. 1877 Gauss: Ein Umriss seines Lebens und Wirkens. Brunswick: Vieweg.

Wittstein, Theodor 1877 Gedächtnissrede auf Carl Friedrich Gauss zur Feier des 30. April 1877. Han over: Hahn.

Zelen, Marvin 1962 Linear Estimation and Related Topics. Pages 558-584 in John Todd (editor), Survey of Numerical Analysis. New York: McGraw-Hill.