Case 3. Let cab; then we have

ax + by = ab

Now as a will divide one member of the equation, it must also divide the other, (PROP. I.); and consequently must divide by. But it cannot divide b, because it is prime to it. Therefore it must divide y, (PROP. III). Hence, ya is its least integral value. Substituting this value in the place of y, we have ax+ab=ab, or ax=0. Wherefore x=0; and the conditions of the equation cannot be fulfilled.

Case 4. Let cab-(ax+by').

Then we have

ax+by=ab(ax+by'), or transposing,

a (x+x)+b (y+y)=ab.

Whence as before, y+y will equal a, for its least value; and xx will become zero. This case is therefore proved.

(To be continued.)

COMPLETE LIST OF DR. BOWDITCH'S WRITINGS.

To the various volumes of the Transactions of the American Academy of Arts and Sciences, Dr. BOWDITCH Communicated the following memoirs:

Vol. II., Part II., Published in 1800.

1. A New Method of Working a Lunar Observation.

The object of this method was to establish a uniform rule for the application of corrections, so that there should be no variation of cases resulting from the distance and altitude of the observed bodies.

Dr. BOWDITCH says of this method, in a note, that "it was written several years ago, and before the publication of the Transactions of the Royal Society for 1797, in which is inserted a method, somewhat similar, invented by Mr. MENDOZA Y RIOS. An appendix to the New Practical Navigator has lately been published, in which the corrections are all additive, and the work is shorter." It is particularly noticed and commended in the Connoissance des Tems (1808) then published under the direction of M. DELAMBRE.*

ZACH (Corr. Astron. Vol. VI., p. 553, A. D. 1822), says: "M. BOWDITCH dans son New American Practical Navigator a aussi donné pour la réduction des distances lunaires une nouvelle méthode VOL. II.

8

2. Observations on the

3. Observations on the [pp. 18-23.]

Vol. III., Part I., Published in 1809.

Comet of 1807. [pp. 1-18.]

Total Eclipse of the Sun, June 16, 1806, made at Salem.

In a note to this communication, Dr. BowDITCH makes, as is believed, the first public mention of an error in LAPLACE'S Mécanique Céleste, in the estimate of the oblateness of the earth, as calculated from the length of pendulums; showing that LAPLACE'S result ought to have been, upon his own principles, 315 instead of 3. 4. Addition to the Memoir on the Solar Eclipse of June 16, 1806. [pp. 23-33]. 5. Application of NAPIER'S Rules for Solving the Cases of Right-Angled Spheric Trigonometry to several Cases of Oblique-Angled Spheric Trigonometry. [pp. 33-38]. This communication so alters NAPIER'S rules, as to make them include most of the cases of oblique-angled spheric trigonometry, and is marked by the same neatness, elegance, and simplicity, which characterized his first communication. These rules are now familiarly known in the text-books of Harvard College as BOWDITCH'S Rules."

Vol. III., Part II., Published in 1815.

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6. An Estimate of the Height, Direction, Velocity, and Magnitude of the Meteor that exploded over Weston, in Connecticut, December 14, 1807. [pp. 213-237]. This communication is of a very interesting character, and it rests upon numerous observations collected with great labor and assiduity. Dr. BOWDITCH considers the meteor in question to have had a course about eighteen miles above the earth, a velocity of more than three miles a second, and a probable cubic bulk of six millions of tons, which others have estimated to be the contents of the pyramid of Cheops.*

7. On the Eclipse of the Sun of September 17, 1811, with the Longitudes of several Places in this Country, deduced from all the Observations of the Eclipses of the Sun, and Transits of Mercury, and Venus, that have been published in the Transactions of the Royal Societies of Paris and London, and the Philosophical Society held at Philadelphia, and the American Academy of Arts and Sciences.† [pp. 255-305.]

abrégée, avec des tables, qui mérite d'être plus connue; aucun auteur Européen n'en a encore parlé; il vient de la perfectionner dans sa quatrième édition stéréotpye publiée à New York en août 1817. Nous la recommandons à l'attention des professeurs et auteurs des traités de navigation." In Vol. X., p. 321, A. D. 1824, he says: "La méthode de M. BOWDITCH a l'avantage sur toutes les autres méthodes d'approximation, que toutes les corrections sout toujours additives, et qu'on n'a jamais besoin de faire attention à des cas particuliers; les règles sout générales ;" and proceeds to give a detailed account of it. See also note to article 15.

*The Zeitschrift für Astronomie, Vol. I., p. 37, A. D. 1816, gives the results arrived at in this communication, and calls it "einer interressanten Arbiet."

t The Zeitschrift für Astronomie, Vol. I., p. 90, 1816, mentions the observations of the eclipses of the sun, June 16, 1806, and September 17, 1811, as contained in these volumes, &c., and states that "BOWDITCH hat den grössern Theil davon zu Längenbestimmungen benutzt und zugleich dabey, für eine Menge Amerikanischer Orte, Hülfsgrössen zur leichtern Berechnung des Nonagesimus gegeben;"

8. Elements of the Orbit of the Comet of 1811. [pp. 313-326].

In this, as in his second communication, he arrives at his results after almost incredible labor, rendered necessary by the want of the improved methods of the present day.* The original volume, containing his calculations in the case of this latter comet, now preserved in his library, contains one hundred and forty-four pages of close figures, probably exceeding one million in number, though the result of this vast labor forms but a communication of twelve pages.†

9. An Estimate of the Height of the White Hills in New Hampshire. [pp. 326-328]. 10. On the Variation of the Magnetic Needle. [pp. 337—344].

This communication, in like manner, which is of quite an interesting character, and of considerable practical importance, was the result of five thousand and twenty-five observations, during a period of five years.

11. On the Motions of a Pendulum suspended from two points. [pp. 413-437].

This communication is also one of interest and value; and the little wooden stand, from which a leaden ball was suspended, still exists, to remind us of the zeal and assiduity with which Dr. BOWDITCH watched the various curves and lines which the ball described.‡

12. A Demonstration of the Rule for finding the Place of a Meteor, in the Second Problem, page 218 of this Volume. [pp. 437-439]

Vol. IV., Part I., Published in 1818.

13. On a Mistake which exists in the Solar Tables of Mayer, Lalande, and Zach. §

and ZACH, in his Corr. Astron. Vol. X., p. 494, A. D. 1824, has a table of the longitudes and latitudes of places determined by astronomical observations calculated by Dr. BOWDITCH.

* See Dr. BowDITCH's letter (ZACH, Corr. Astron. Vol. X., p. 228), before referred to, where this fact is stated. The editor, in page 248, gives the elements of the orbits of the comets calculated by Dr. BOWDITCH Wholly from American Observations.

† Mr. ENCKE, in speaking to a friend of Dr. BOWDITCH, at Berlin, in 1836, said that he had known him from the time when this paper appeared; and that he had never seen an American since, without asking him what he could tell him about its author; and the Zeitschrift für Astronomie, Vol. I., p. 44, gives an account of this communication "von dem Amerikanischen Astronomen BOWDITCH." This subject is mentioned is his letter to Baron ZACH, before alluded to, (Corr. Astron. Vol. X., p. 227). The editor, in his note, p. 246, says the remarkable variety of the motions of a pendulum thus suspended, and the very curious experiments of Prof. DEAN, who explains, in this mode, the apparent motion of the earth as seen from the moon, engaged Dr. BoWDITCH in the examination of the theory of these motions. The result has been, he adds, "une recherche très intéressante." "Comme ce mémoire mérite d'être mieux connu, et qu'il ne l'est pas généralement, vu la difficulté de se procurer des livres Américains, nous en donnerons la traduction dans un de nos cahiers."

§ Dr. BOWDITCH states, that "The attraction of Jupiter produces an equation in the expression of the Sun's distance from the earth, and a Table is given for its computation, by MAYER, in 1770,” &c.," and ever since this table was first published, which is about fifty years, an error of six signs has always existed in the argument, by which the correction is found; so that, when the equation is really subtractive, it will frequently be found by the table to be additive, and the contrary." "In DE LAM

14. On the Calculation of the Oblateness of the Earth, by means of the observed Lengths of a Pendulum in different Latitudes, according to the Method given by Laplace, in the Second Volume of his "Mécanique Céleste;" with remarks on other parts of the same Work relating to the Figure of the Earth. [pp. 3-24.] The object of this communication is to correct certain errors in the article "Earth," in REE's Cyclopædia, to the end that currency should not be given to inaccurate ideas on the subject, by that popular work.

15. Method of Correcting the apparent Distance of the Moon from the Sun, or a Star, for the Effects of Parallax, and Refraction. [pp. 24-31.]

This is but the rule given in the Practical Navigator, making all the corrections in question additive. It is another instance of the simplicity at which he always aimed in his rules and formulas.*

16. On the Method of computing the dip of the Magnetic Needle in different Latitudes, according to the Theory of Mr. Biot. [pp. 31-36.]

17. Remarks on the Methods of correcting the Elements of the Orbit of a Comet, in Newton's "Principia" and in Laplace's "Mécanique Céleste. [pp. 36-48.]

This communication proves that two equations in the Principia, the accuracy of which several commentators upon that work had attempted to prove, and as to which no doubts had yet been expressed or insinuated, always made the corrections in question "double of what they ought to be,” and restricts the method of LAPLACE as appropriate only where the number of observations is small.

18. Remarks on the usual Demonstration of the Permanency of the Solar System, with respect to the Eccentricities and Inclinations of the Orbits of the Planets. [pp. 48-51.]

19. Remarks on Dr. Stewart's Formula for computing the Motion of the Moon's Apsides, as given in the Supplement to the Encyclopædia Britannica. [pp. 51-61.]

This is a very curious and interesting communication. A method which, notwithstanding doubts had been expressed respecting it, had been sanctioned as accurate by

BRE's Solar Tables, published in 1806, the form of the table is wholly altered, the method of entry by a double argument being used; and by thus taking a different path, the error is avoided, without noticing that it really does exist in the other works."

Baron ZACH, in his Monatliche Correspondenz, Vol. VIII., p. 449, A. D. 1803, says that BowDITCH, an American astronomer, has called attention to this mistake; and, after admitting its importance, frankly adds, "Allen Astronomen, welche sich mit Verfertigung der Sonnen-Taflen beschäftigt haben, einen LA CAILLE, TOB. MAYER, LA LANDE, DE LAMBRE und mir ist dieser Fehler entgangen." * In ZACH's Monatl. Corres., Vol. XVII., p. 411, A. D., 1808, this method is mentioned as being in the Appendix to the New American Practical Navigator, printed in Newburyport, 1804; and the editor says: "Der Verfasser ist ein Americaner, BOWDITCH, und DE LAMBRE hat es der Mühe werth gehalten eine umständliche Darstellung dieses Verfahrens zu geben." Then follows a somewhat minute account of the method. See note to Article I.

Dr. HUTTON, by LALANDE, and PLAYFAIR,-the latter of whom even considered its accuracy to have been demonstrated, -is in this memoir proved to have been true only in the particular case supposed; and it is shown that, as a general method, it wholly fails.

Vol. IV., Part II., Published in 1820.

20. On the Meteor which passed over Wilmington, in the State of Delaware, November 21, 1819. [pp. 3-14.]

21. Occultation of Spica by the Moon, observed at Salem. [p. 14.]

22. On a Mistake which occurs in the Calculation of Mr. Poisson relative to the Distribution of the Electrical Matter upon the Surfaces of two Globes, in Vol. XII., of the Mémoires de la classe des sciences mathématiques et physiques de l' Institut Impérial de France. [pp. 15-17.]

23. *Elements of the Comet of 1819. [pp. 17-19.]

Besides the above contributions to the Memoirs of the American Academy, Dr. BOWDITCH was the writer of several other articles, among which may be mentioned the following:

1. Notice of the Comet of 1807. Published in the Monthly Anthology, for December, 1807, Vol. IV. [pp. 653-654.]

2. Review of a "Report of the Committee (of Congress) to whom was referred, on the 25th January, 1810, the Memorial of William Lambert, accompanied with sundry Papers relating to the establishment of a First Meridian for the United States, at the permanent seat of their Government. Published in the Monthly Anthology, for October, 1810, Vol. IX. [pp. 245–266.]

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This article occupies twenty-one pages, and proves very conclusively the great advantages of continuing to estimate the longitude from Greenwich, which Mr. LAMBERT considered "a sort of degrading and unnecessary dependence on a foreign nation," and an incumbrance unworthy of the freedom and sovereignty of the American people." This Memorial the reviewer shows to be a compilation, with needless repetitions and palpable mistakes, evincing a great want of knowledge in the principles of the calculations ;" and that, "both as respects its object and execution, it was wholly undeserving the patronage of the National Legislature."

3. Defence of the Review of Mr. Lambert's Memorial. Published in the Monthly Anthology, for January, 1811, Vol. X. [pp. 40-49.]

16

Mr. LAMBERT having made an angry reply, charging his reviewer with "twistical cunning," ingenious quibbling," "zeal for the honor of the British nation, and convenience of British mariners," and challenging him "to examine his computation of the longitude of the Capitol at Washington from Greenwich, and to point out a mistake that

* For a statement of Dr. BOWDITCH's communications to the Memoirs of the Academy, and an abstract of their contents, from which several of our remarks in the text are condensed, see Mr. PICKERING'S Eulogy, [pp. 17-31].