longitude of the nonagesimal of some particular place. In other words, the rule given by Mr. Lambert leads to this absurd conclusion, that the moon's orbit is not inclined to the ecliptick! Whence the incorrectness of the method is sufficiently obvious. We have preferred taking this simple example, in order to point out the defects of the formula, rather than enter into a minute discussion of it, particularly as the thing is already done to our hands in the works of La Lande, Vince and Willard, mentioned in the preceding note.

Again. Mr. Lambert observes, "I have stated in my paper No. 2, that the parallactic angle-is found by either adding the angle of position to, or subtracting it from, the angle between the meridian passing through the centre of the sun, moon or star, and a parallel to the ecliptic-and this in your opinion is one of my palpable mistakes. You ought to be careful not to suffer your great anxiety to find out imperfections in others, to lead to the discovery and exposure of your own." What Mr. Lambert could mean by making this assertion as to our opinion on this point, we leave to the reader to judge. If he will turn to our Review, in page 262, he will find that the only mention we have made of the rule for finding the "parallactic angle" is, that the part of it for finding the angle between the parallels of the ecliptick and equator (which is equal to the angle of position as defined by most astronomers) is the same as in 1047 of La Lande's or ¶754 of Vince's astronomy; and we have never expressed the least doubt of the accuracy of the rule in any part of our review. On this point Mr. Lambert must therefore rest under the imputation of gross ignorance or wilful perversion of the truth. Perhaps it was through ignorance in not understanding the import of the words that followed in page 262, viz. " and the rule for applying the parallax in latitude is not correct in some cases where the moon is between the zenith and the elevated pole of the ecliptick." If Mr. Lambert does not understand this, we will explain it to him by an example. Suppose that at a place between the tropicks in north latitude, the moon was in the same longitude as the nonagesimal and between the zenith and the north pole of the ecliptick, (or between the points corresponding to Z, E) the parallax in latitude would, in this case, decrease the north polar distance, and if the moon's latitude were north, the parallax ought evidently to be added to the true latitude instead of subtracted, as directed by Mr. Lambert in his

This we considered as one

rule given in the preceding note. of his palpable mistakes. If he wishes us to point out a few more, we will mention his method of finding the Arch II in the rules 2, 3 page 12, where he directs in all cases to subtract the Arch I, whereas it ought to be added in some cases where the horary angle exceeds 90°. We might mention many other instances.

Mr. Lambert, unable to account for the difference between the ratio of the equatorial diameter and the polar axis of the earth as calculated by Newton and La Place, intimated in his paper that it might be owing to "a diminution in the equatorial diameter." We stated the various causes which made astronomers assume a different ratio from that of Newton, and pointed out a mistake of Mr. Lambert in asserting, "that in consequence of new lunar equations, discovered in France by Mr. de La Place, the proportion of the equatorial diameter to the polar axis of the earth is now assumed as 334 to 333." Mr. Lambert endeavours to evade the question, and insinuates that there is an errour in the calculation of La Place of the ellipticity (deduced from combining the degrees of the meridian in France and Peru) or in that of (deduced from the lunar equations) which are the only ratios of La Place that we have used in our review. Now we state explicitly that both these calculations are perfectly correct, upon the principles and data assumed by La Place*, and we challenge Mr. Lambert to prove the contrary. It is true that La Place has in one instance made a small mistake in calculating the ellipticity 78 from the lengths of pendulums in different latitudes; and the ratio from combining all the degrees of the meridian measured in different parts of the earth, requires some modification on account of the late measurement in Sweden; but neither of these calculations have any reference to the ratio used by Mr. Lambert, on which our former remarks were founded.

Mr. Lambert, in another part of his answer, says, "I demand, in explicit and unequivocal terms, of you, and your mathematical coadjutors, (if any you have) to examine the computation of the longitude of the capitol in the city of

* La Place generally neglects terms of the same order aş the square of the ellipticity, as is usual in such calculations.

Washington from Greenwich, on the principles and data contained in my paper No. 2, and point out a mistake that you can make palpable in the result. This is coming at once to the point in question; and let us have no more of your ingenious quibblings, sophistical evasions and subtle prevarications, so far as they relate to my plan, its execution or object."

We are willing to gratify Mr. Lambert in his request, and shall therefore answer "in explicit and unequivocal terms," that there is an errour in every one of the six examples he has given in page 14, in calculating the parallax in latitude; his estimate being too great in every instance by about five seconds; and this errour affects the calculation in the following pages, though we presume that the estimated longitude of the place will not be materially altered, since the errour of the parallax at the emersion has a tendency to balance the errour at the immersion, and thus, by one of his errours correcting the other, the result may be nearly correct. If this is not explicit enough, we will also add, that the cause of this mistake is his having erroneously introduced the term cosine of the moon's latitude in his rule page 13, and not having introduced it in his rule page 11*. This is a fair sample of that kind of mistake of which Mr. Lambert complains that we accuse him-of sometimes doing too much, and sometimes too little.

Mr. Lambert pretends to compare the proposed change of meridians to that made in the currencies of this country by the introduction of the decimal ratio of dollars, dimes, cents and mills. But the cases are not parallel; so far, however, as Mr. Lambert's example is applicable, it is against his argument; for by the former change four different currencies were reduced to one uniform method of computation, by which all the calculations were much facilitated; whereas by Mr. Lambert's project, instead of one meridian, we shall be

* As Mr. Lambert insists that his rule is correct, notwithstanding the authority of La Lande to the contrary, we shall mention another authority that we presume will not be disputed by Mr. Lambert, namely, that of Mr. Garnett, in page 48 of the appendix to his Requisite Tables, where the rule is given correctly, and we would ask what has become of the term cosine / in the value of sine 2, which term is neglected in the rule page 11.

obliged to make use of two, to the great embarrassment of all geographical and nautical calculations.

We need not spend time in examining Mr. Lambert's opinion, that the establishment of a first meridian is as much the prerogative and evidence of sovereignty as the establishment of a mint, or the forming a standard of weights and measures. But we may remark that the business of fixing a first meridian is generally left to astronomers and geographers to manage in their own way. Every citizen, in his intercourse with others, is under the necessity of making use of the coins, weights and measures established by the laws of his country; but the case is essentially different with respect to a first meridian, since each individual has it completely in his power to use any one he pleases.

Having, in our former review, pointed out the sources from which Mr. Lambert had compiled most of his rules, the few that remained unnoticed seemed hardly worth his claiming; but Mr. Lambert has formed quite a different opinion, and says, "I have constructed two tables of logarithms on a new plan and accurate principles, in my papers No. 4 and 5, and given a new rule in No. 3, to find the moon's hourly velocity, but you will not agree that I am entitled to the least credit for them." As Mr. Lambert appears to rest his chief claim to originality on these points, we shall briefly examine them.

The Table in No. 5, for finding the moon's horizontal parallax for any latitude is useful, but it is not on a new plan. For one exactly similar was published in Burg's tables (tab. XLV.) a long time before Mr. Lambert's paper*. Table No. 4, for finding the augmentation of the moon's semidiameter on account of the elevation above the horizon may be on a new plan, but it is a plan we believe that no one except the author will follow; since the object may be obtained in less than half the time, and in a way that is less liable to any great mistake, by a table of double entry like Table XLIV of Burg, so that on this score the publick are not under any great obligation to Mr. Lambert. The rule for calculating the moon's hourly

* These tables were published by the board of longitude in France in 1806, and republished in England in Vince's Astronomy, vol. 3, in 1808. It is this last edition to which we have referred.

velocity when the time from noon or midnight is not an aliquot part of 12 hours, or a fractional part expressed in small numbers, is far more laborious than Mr. Garnett's method, (given by Mr. Lambert) and as a considerable number of solutions of this question may be found by any tyro in mathematicks, by combining in various ways the differences of the several orders, we think but little merit can be attached to the discovery of one formula of this kind, unless it be more simple than those in general use; and as Mr. Lambert's method is of an opposite kind, his claim to originality in this instance, as well as in the others, does not deserve much attention.

There are many other points of minor consideration in Mr. Lambert's answer, in which we might point out gross errours and mis-statements, but as we have already extended our remarks to a great length, we shall refrain.

We have now examined the most important parts of the defence of Mr. Lambert's rules and calculations, and proved him to be wrong in every instance, and even in the calculation of the occultation at Washington, in which he more particularly challenged us to discover one single mistake—we have pointed out several. In doing this, we have been influenced not so much by a desire of vindicating ourselves from his impotent aspersions, as of lending our aid (so far as we are able) to the cause of truth. Before closing our remarks, however, we shall observe, that the whole tenour of Mr. Lambert's papers bears strong marks of his wishing to make the business of his memorial a question of party politicks, than which nothing can be more improper and unworthy a man of real science. We trust that the good sense of the legislature of the United States, will prevent the adoption of a scheme that would be so injurious to the cause of science in our country.