the positions, and divide by the difference of the results; the quotient is a correction to be added to the first position, if the first result shows it to be too small, and to be subtracted from the first position if the first result shows it to be too large.

This rule we prefer to the old rule quoted from DABOLL, although it gives, of course, the same result.

ON THE INDETERMINATE ANALYSIS.

By Rev. A. D. WHEELER, Brunswick, Maine.

[Continued from Page 25.]

PROPOSITION VI. If a and b be prime to each other, the indeterminate equation, ax — -by=c, is always possible; and will admit of an infinite number of positive integral solutions.

DEMONSTRATION. Transferring and dividing, we have x= which has already been shown to be possible. PROP. V., Cor. 2.

c+by

a

Now as the remainders, resulting from the division of c + by by a, recur in periods, and the number of periods is unlimited, since y may have all possible values, it follows that the number of solutions must also be unlimited; that is, infinite.

PROP. VII. If, in the equation ax + by=c, we call the least integral value of x,v; the next greater value of x, when the equation admits of more than one solution, will be vb, the next v+2b, and so on in Arithmetical progression.

DEM. The least value of x being v, and d denoting the difference, we shall have v+d for the next greatest. Then ax + by becomes av by' c in the first case, or y

=

c-av

In the second case, it becomes av+ad ± by"=c, or y′′

an integer.

=

cav

c-av-ad

±b

c-av-ad

ad ±b'

(Ax. 2). Therefore

d

is an integer. (PROP. III.) But this can be

an integer only when d=b, d=26, &c., which was to be proved. PROP. VIII. The equation ax + byc is always possible for n solutions, when c> n a b.

DEM. Let cnab+r. Then we have

ax+by=nab+r; or

ax-nab――by+r; or

(PROP. V. Cor. 1). Therefore the equation admits of at least one solution.

Let x=v for its first value; x=v+b for its second; x=v+ 26 for its third; and so on, (PROP. VII). Then we shall have x= v+(n−1)b, for its nth value; and substituting this for x, we have av + (n-1) ab+by=nab+r; or

Therefore the equation ax+by=c, will always admit of n solutions when c> nab.

PROP. IX. The equation ax + byc is impossible, in positive whole numbers, in the following cases. (1.) When a, or b, is prime to c, but not prime to each other. (2.) When c <a+b. (3.) When cab. (4.) When c = ab · (ax+by').

DEM. Case 1. Since a and b are supposed to have a common factor which is not in c, it is obvious that one member of the equation is divisible by it, while the other is not.

Case 2. The smallest integral value which can be given to x or y is 1. Giving to them this value, the equation becomes a+b= c. Consequently if c is less than a + b, the solution is impossible.

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

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, 15 instead of 6. 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.

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."

†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