(Translated from the Rivista Marittima Italiana by F. W. Morrison.)
1. The triangle of position may be solved without trigonometrical computation in three ways:
(a) By numerical tables.
(b) By abacuses.
(c) By geometrical instruments.
2. Attempts of the first sort are very numerous and may be classified in groups according to the problem to be solved. The most important groups are those dealing with the calculation of the altitude, that of the hour angle, and that of the azimuth.
As the number of such attempts is constantly increasing1 we
1 The most recent attempts at the calculation of the altitude are: That of Ball (Altitude Tables, London, 1908, Potter), and that of Guyou (Nouvelle
Mithode pour determiner le point a la mer. Comptes rendus, 1908, No. 21).
Ball's large table is one of three entries and gives the altitude directly
(to tenths of a minute) with the latitude, hour angle and declination (these three elements to the nearest degree) ; in our opinion, however, they present two inconveniences: that of serving only for values of d included between — 24° and 24°, and that of substituting for the position by dead reckoning an assumed position (in order to avoid interpolations of latitude and hour angle) whose distance from the former may be four times as great as Souillagouet thinks permissible. (Ball's 3d volume, ö and L to 60°, has now been published.—P. R. A.)
Guyou's new method also, although theoretically very attractive, does not seem to be without its drawbacks, from a practical point of view, since it requires the use of two double entry tables (one of which is the table of Lord Kelvin), and strict attention to the signs, because, for example, the auxiliary declination and also the auxiliary altitude may have signs contrary to those of the true declination and the true altitude.
But, in this connection, we may again be permitted to insist upon the conclusions at which we have already arrived in our papers Sul calcolo relativo alle rette d'altezza (Rio. Mont., Jan., 1903, and April, 1904). It seems to us that all the methods, more or less indirect, proposed for the calculation of the altitude are generally unequal to the purpose, and that the simplest method is always that offered directly by trigonometry and based upon the use of Euler's formula transformed by prosthaphxresis (see See. 5 of the second of the above-cited papers), for which a table of natural cosines with one minute intervals and calculated to five places of decimals is more than sufficient; and it is with great pleasure that we have seen this our opinion shared by Lieut. Segre (Rio. Marit., appendix to the issue of March, 1909.)
Furthermore, we remember that the one who proposed (at the beginning of the last century) the use of natural values for all the principal nautical calculations, was Francisco de Paula Travassos; see his Methodo de reduced das distancias observadas (Coimbra, Real Imprensa, 1805) to which is annexed a table of natural cosines only, which, having then to serve also for the calculation of lunar distances, is calculated for every 10” and to six places of decimals.
believe it would be useful to publish critical comparative studies (such as we have already published)2 of the various special tables belonging to each group.
Such studies would be extremely useful to those who desire to simplify, for a special reason, the solution of some fundamental problem, without a knowledge of the bibliography of the subject; indeed, in not a few cases, these studies would show that the problem they had set themselves was already solved or impossible of solution.
These studies would also prove valuable by showing, in the case of each group, what special tables are preferable. The advantage of such a knowledge might, perhaps, not be felt by such as have already become accustomed to some fixed method, for it is the method already known that is usually held to be the
2 On Lord Kelvin's Tables for Facilitating Sumner's Method at Sea. (Rio. Mont., Jan., 1909).
Since Dr. Pesci's article was written there have been published "Altitude and Azimuth Tables for Facilitating the Determination of Lines of Position and Geographical Position at Sea," by Lieutenant Radler de Aquino, Brazilian Navy. These tables were described by their author in the December, 1908, number of the U. S. NAVAL INSTITUTE PROCEEDINGS, and may be obtained from their publisher, J. D. Potter, 145 Minories, London (10s. 6d.), or from Messrs. John Bliss & Co., Front Street, New York.—P. R. A.
easiest and the quickest; it would be, nevertheless, most important for those beginning the study of nautical calculations. 3
3. Critical comparative studies of the kind indicated should be made, chiefly, from two points of view: having regard to practicability and to the degree of approximation.
The practical value of tables depends upon the rapidity and facility with which they can be used.
For the purpose of rapidity a small table is frequently preferred to a large one, though we believe unadvisedly, for the former undoubtedly requires more time than the latter, on account of interpolations. There are, in fact, examples of this, such as the well-known traverse tables and those for calculating the tides.
As for facility of use, tables should have for their arguments of entry the data. Examples of this are the azimuth tables of Bolte4 and Fulst. 5
In both these methods a numerical value is first calculated by means of two special tables, and by means of this value, the azimuth is obtained from a third table; but in the third table of Bolte, the azimuth is an argument of entry, while in that of
Fulst the azimuth is a value given by the table itself. For the above reason, Fulst's tables (which we believe could advantageously be made more extensive) are to be preferred to those of Bolte.
4. Finding the degree of approximation constitutes a problem of considerably more difficulty.
For tables of only one entry, formulae are available (deduced from the theory of interpolations) that give, for the errors of interpolation, upper limits very close to the errors themselves. 6
3 ”it is customary to cling to old ways, though it would be better to be somewhat more sceptical of their efficacy. If one is not sufficiently enlightened to abandon them, it would be advisable at least not to transmit them to one's pupils, to whom it is possible to show better ones." Delambre, History of Astronomy, Vol. I, P. 264.
4 Tavole azirnutali. Authorized translation by A. Vital. Trieste, Schimpff, 1900.
5 Azimut-Tafel. Bremen, Heinsius Nachfolger, 1898.
6 These formulae applied to ordinary trigonometrical log, tables, give such narrow limits that they could never be made sensibly less (for ordinary purposes) whatever other method be followed in finding them. See our two articles: Sulla recerca del logaritmo-seno e del logaritmo-tangente degli arc/ti piccoli; La Corrispondenza, 1901; Periodic di Matematica, Igor; Sulle operazioni fra numeri decimali approsimati, Periodico di Matematica, 1904.
But for tables of double and triple entry, formula that can be deduced from Taylor's theorem are the only ones available up to the present time; and these give, generally, extreme limits too far from the errors under consideration; it is therefore impossible to get from them the criteria necessary for limiting the differences between the arguments of entry.
Hence it is necessary to consider only one of the arguments as variable, and then to study the errors of interpolation for each of the values assigned the other arguments: a laborious task, that might, however, be rendered less arduous by utilizing the principle of continuity. For example, in the group of tables giving the hour angle, those of Davis7 are very well known. Now from these, for ?=48° 30', ?=0° 30', h=36° 30', we have P with an error greater than 8'; while for ?=49° 30', ? = —22° 30', h= 16° 45', we have P with an error greater than 24'. It is quite true that these cases are somewhat unfavorable, but where do these cases begin, and why are they not excluded?
5. In a recent and authoritative bibliography appearing in the Rivista Marittima8 the question is asked whether it would not be advisable to construct a general table9 for the solution of the triangle of position. Such a table, carefully planned in its arrangement and its limits, would do away with a great many special tables (an unquestionable advantage).
We believe it interesting, in this connection, to set down here a noteworthy statement of Lord Kelvin. 10
When we consider the thousands of triangles daily calculated on all the ships at sea, we might be led for a moment to imagine that every one has already been solved, and that each new calculation is merely a repetition of one already made; but this would be a prodigious error, for nothing short of accuracy to the nearest minute in the use of data would thoroughly suffice for practical purposes. Now there are 5400 minutes in 90°, and
7 Chronometer Tables. London, Potter, 1899.
8 By Doctor Alessio, on the Position Line Star Tables, of Goodwin (March, 1907).
9 Thomson's Table, examined in our article cited in Sec. 2, may be considered as a general table for the solution of the spherical right triangle, but not for the direct solution of all spherical triangles.
10 Proceedings of the Royal Society (of London), Vol. XXIX, 1870-71, p. 260. This statement is repeated, with a different wording, in The Tables for 'Facilitating, etc. therefore there are 54003 or 157,464,000,000 triangles, to be solved for a single angle. This, at moo fresh triangles per day, would occupy about 400,000 years. Even with an artifice such as that to be described below, for utilizing solutions of triangles whose sides are integral numbers of degrees, the number to be solved (being 908 or 729,000) would be too great, and the tabulation of the solutions would be too complicated.
Three objections, however, may be made to this statement. First, it is not enough to assume that all three sides are less than 90 degrees, for where the latitude and the declination are of contrary name, the triangle has one side greater than 90 degrees, and its general solution cannot (without trigonometrical calculation) be reduced to the solution of a triangle with all three sides less than 90 degrees. The second objection is that in solving all the spherical triangles possible (given the three sides), it is not necessary to consider the permutations of these sides, as has evidently been done to obtain the above results. It is enough to consider the combinations, since in varying the order of data, a new calculation does not arise and it will suffice to make a corresponding variation in the order of the angles. The third objection is that before making useless calculations it would be determined a priori whether the conditions necessary for the existence of the triangle are satisfied.
We have, in view of these objections, made another investigation, and by a rather arduous process of reasoning, which we do not think necessary to record here,' we are able to show that the triangles to be solved are 26,258,581,800, if the sides are considered to vary by 1’: or 125,580 if the sides vary by 1°: in either case the triangles to be solved are less than one-sixth as many as those indicated by Thomson.
The first result has no practical importance, and serves merely to satisfy a purely scientific curiosity. This cannot be said, however, of the second, and contrary to the conclusion reached by Lord Kelvin (a conclusion based on a result six times greater than that found by us) we are of the opinion that a table in which the sides vary by whole degrees could easily be calculated and included in a single volume. We believe, too, that there could also be added to this table the three differences for I', given by Davis, which simplify considerably the interpolations. This general table would be of the utmost utility by doing away with many
11 Un Problema d’analisi combinatorio. Periodico di Matem., 1908, Vol. XXIII, fasc. IV.
special tables (chronometer tables, azimuth tables, etc.) used in nautical calculations. We repeat, however, that such a table should be most carefully planned as to its arrangements and limitations.
6. Coming to the second type (cf. Sec. I): Methods for the nomographical solution of special problems are numerous. We believe the most important to be:
Nomograms of the Sun's Azimuth, constructed by Professor Molfino, (Annali Idrografici, 1901), and by Lieutenant Perret, of the French Navy (Note sur quelques applications de la Nomographie, Annales Hydrographiques, 1904).
Abacus for the Calculation of Latitude by Circumeridian Altitudes, constructed by the author of this paper (Riv. Marit., 1898) ; with a diagram 45 x 35 cm., the result is obtained with a more than sufficient approximation.
Nomogram for the Calculation of Corresponding Altitudes, constructed by Lieutenant Perret (Note sur la construction d'un nomogram. Paris, Librairie Chapelot, 1905). In our own article (Sul calcolo della retta d'altezza. Riv. Marit., Jan., 1903) we alluded to the construction of this and other abacuses; we would now add that with an abacus closely resembling the one referred to, it would be possible to obtain Pagel's coefficient as a function of declination, altitude and hour angle. 12
7. Numerous attempts have also been made at the general solution of a spherical triangle, the two most noteworthy of which are certainly those of D'Ocagne. 13 All of them, however, offer certain disadvantages for the calculations of nautical astronomy, as, for example, the requiring of almost impracticable dimensions for an approximation itself hardly adequate.
Abacuses have also been used for the solution of a spherical right triangle (to which, as we stated in our article on Lord Kelvin's tables, may be reduced the solution of any triangle whatsoever). Two of these abacuses have been constructed with the dimensions required for actual use.
12 In our article, Las Tablas grdficas de Luyando, Contribucion d la Historia de la Nontagraphia, Annals of the Faculty of Sciences of Zaragoza, September, 1908, March, 1909 is to be found a catalogue of all the abacuses used in Navigation (known to us) up to March, 1906.
Pagel's co-efficient is the change of hour angle for a change of one minute of arc in the latitude.—P. R. A.
13 Traite de Nomographie, Secs. 123 and 124. Paris, Gauthiers-Villars, 1899.
The first is that of Messrs. Fave and Rollet de L'Isle : 14 its dimensions are not practical (about a square meter), and it does not always give a sufficient degree of approximation, as we have demonstrated in our article, Resolucdo Nomograpkica do Triangolo de Posicao. 15
The second is that of Mr. G. W. Littlehales, hydrographic engineer in the United States, whose name was already well known to us. 16 This abacus17 is based, not on any nomographical principle, but on a very ingenious artifice which we analyzed in the article already mentioned. This abacus has the serious defect of occupying an area of 14.60 m. (which could, however, be reduced to one-fourth), divided among 365 pages. The minor defects are these: it is not neatly designed, it is not easily legible, and it is expensive ($25.00 in Philadelphia). 18
In short, an abacus for the solution of any spherical triangle,
14 A model (36 cm. X 36 cm.) of this abacus was recently published by Prof. Constan, author of the well-known Cours Elementaire d'Astronomie et de Navigation, under the title, Tables graphiques d'azimut. Saint-Brienc, Guyon, 1936.
15 Rivista Maritima Brazileira, November and December, 1907, and February, 1908.
16 See Pilot Chart of the North Atlantic Ocean. January, 1903.
17 Simplest Method of Navigation. Philadelphia, Lippincott Co., 1903.
18 We should add to these two diagrams that of Chauvenet : Diagram for the Solution of the Spherical Triangle, according to Chauvenet, recently edited by Dr. L. Kohlschutter, published by Dietrich Reimer, Berlin, 1905, as well as the Azimuth Diagram of Dr. Alessio, Riv. Marit., supplement to the July number, 1908.
Chauvenet's Diagram consists of a stereographic projection placed on another like projection and revolving around a common center. This diagram, we must say, seems to us hardly practical (I) because its dimensions are too small (the degree of approximation attainable is little less than a degree), (2) because of the defacement of the transparent material on which the movable part is traced, which reduced still more the degree of approximation. In the copy examined by us the results are altered more than half by this defacement.
The diagram of Signor Alessio is a polar abacus (see Nomography, p. 119), which could be reconstructed from a Fave abacus by a simple graphic process (see Notes on Nomography, Riv. Marit., August, 1889, Sec. 19). It consists of two scenographic projections, one meridian and the other polar, and placed one over the other. We believe that its construction would be simplified considerably by placing an equidistant meridian projection over a polar projection also equidistant, since the latter is simple and uniform in design and does not require numerical calculation. Over the latter may be traced (also without calculation) the other projection by using Table III of Souillagouet. But once the diagram is made, this consideration has no practical importance.
Coming to the use of the diagram itself, we, too, are of the opinion that it may be useful for problems in which an approximation of 15' at most is required (see the following Sec. 8) ; it would be desirable, however, to have it reduced to one part, thereby making it less cumbersome and doing away with the necessity of transferring the alidade from one part to another. This, however, would be equivalent to considering all the elements as positive and reduced to the prime quadrant and the result would be, either two groups of rulers like those given in another connection by Dr. Alessio (pp. 58 and 69 of the article cited) or mirrors like those given in our article, on The Tables, etc., already cited. This, says the author (at the bottom of page r5), would constitute a greater disadvantage than those mentioned above.
or any right triangle, which is at the same time practical and productive of a sufficient degree of approximation for the calculations of nautical astronomy, does not yet exist, and it is to be feared that, with our present state of knowledge, it is not possible to construct one. 19
8. It now remains for us to consider the instruments.
In the applications of plain trigonometry the degree of approximation required is often greatly limited. Consequently it is easy to conceive an instrument, which, in the solution of plane triangles, might be useful in many cases, and there are well known examples of this. But this would not apply to spherical trigonometry, where the cases are rare in which only a limited degree of approximation (for example 15') is required.
In nautical astrontomy the problems in which this occurs are almost exclusively confined to (I) finding an azimuth, (2) finding the name of an observed star, and (3) calculation of the times of rising, setting, and passage across the prime vertical.
For these few cases and analogous ones (such as those refer
19 Nevertheless, in the second part of our article already quoted (Resoluck), etc.), we explained an experiment of our own in the construction of a general abacus: its construction is quite simple in that it consists of two abacuses, one lying on the other. This abacus might be made practical and capable of an approximation generally sufficient. One of the abacuses referred to has already been reproduced by de Aquino, Nomograms for Deducing Altitude and Azimuth, U. S. NAVAL INSTITUTE PROCEEDINGS, No. 126 (see also Muir, Navigation and Compass Deviations, published by the U. S. Naval Institute. Annapolis, Md., U. S. A. Price, $5.00, gold).
ring to orthodromic navigation), we have the well-known Navisphere of Aved de Magnac.20
A solution such as the above has also been suggested for ordinary problems, in which the degree of approximation must be greater;21 we do not, however, find the proposition practical, principally because it requires that the triangle shall be most accurately traced upon a sphere, and then that there be made upon it very accurate readings.
Another instrument has been proposed for the solution of two of the fundamental cases (given three sides, and given two sides and the included angle) when an approximation of 20' or 30' is sufficient. This instrument is the Azimuthometer of Lieutenant Segre.22 With this instrument the trihedron (given the three faces) may be solved as indicated by descriptive geometry; moreover its author has ingeniously found a way of solving with this instrument the second of the two cases mentioned.
We will close this brief review by examining, somewhat more extensively, an instrument for the solution of spherical triangles invented more than a century ago by the engineer Jean Francisco Richer. 23
The Academy of Sciences of Paris announced in 1790 a prize competition for essays on the following question: "To find a sure and accurate method for the reduction of the apparent distance of two celestial bodies to the true distance, requiring in practice only the simplest calculations within the reach of the great majority of navigators." But, "among the essays submitted there was not one that offered a sufficiently true and accurate solution of the problem," and the contest was postponed to the following year. "Inasmuch as most of the competitors were merely artisans who had done their best to invent instruments which would effect the necessary reduction, citizen Lagrange," fearing "lest they should make further efforts with no greater success, set about to discover some principle that would afford a basis for the construction of such instruments."
20 Heath & Co.: L. Crayford, London.
21 Prof. E. /pPollto, Uso del globo celeste. Rassegna Navale, 1894.
22 Per la rettifica della deviazione magnetica in mare. Riv. Marit., November, 1903.
23 Supplement a la Trigonometric spherique et a la Navigation de Bezont, by Francois Callet. Paris, Didot, An. VI. (1798.)
As a result the distinguished mathematician demonstrated that Euler's equation
cos a=cos b cos c + sin b sin c cos a
may be transformed into
4 sin2 a/2 = (sin b+c/2 + sin b-c/2) 2 + (sin b+c/2 - sin b-c/2) 2
-2(sin b+c/2 + sin b-c/2) (sin b+c/2 - sin b-c/2) cos a, (21)
but this equation is merely Carnot's formula applied to a plane triangle having sides respectively equal to sin b+c/2 + sin b-c/2 and sin b+c/2-sin b-c/2, the included angle equal to a and the opposite side equal to 2 sin a/2; it is therefore sufficient to imagine an instrument by which this triangle may be readily constructed, since,
given the sides b and c and one of the two parts a or a, you have at once the other part. And, "it appears that citizen. Richer was the most impressed by this, for the instrument that won the prize in 1791 was one he constructed on this principle."
We shall explain briefly the construction of this instrument, named by its inventor the Trigonometric Compass. It should be noticed that the scales (which in the figure appended are not indicated), as well as the description, refers here, not only to the solution of the triangle of position, but to that of any spherical triangle.
10. This instrument is composed of two equal principal arms ABB'A', ACC"A" (see diagram appended) hinged at A, in each of which is dove-tailed a movable auxiliary arm, ED and PG respectively. In addition, hinged at B, at the end of the first principal arm, there is a third auxiliary arm BH., which by means of simple connections, always remains in contact with the end C of the second principal arm. Finally there is a fourth auxiliary arm OK, hinged at the end E of the auxiliary arm ED, the center O being a prolongation of the line BA.
Each of the two principal arms is graduated on the edge of the groove indicated in the diagram, according to the laws of sines (with their common length as a unit), commencing at AA' and AA", but the divisions indicate twice the measure, instead of the measure itself, of the corresponding arc. The two auxiliary arms ED and FG are graduated in the same manner, the first commencing at E, and the second at F; to the latter is attached an index L, which in a direction parallel to AA" carries the
origin 0° of its scale FG along the edge AC. Finally, the two arms OK and BH are also graduated, commencing at O and B, like the four preceding ones except that their divisions are twice as great.
From the above it is clear:
1st. If the graduation in of the arm AB coincides with the graduation n of the arm ED, the result is
AO=sin m/2 + sin n/2.
2d. If the graduation in of the arm AC coincides with the graduation n of the arm FG, the result is
AL=sin m/2 - sin n/2.
3d. If the graduation p of the arm OK coincides with the index L of the arm FG, the result is
OL= 2 sin p/2.
4th. If the graduation q of the arm BH coincides with the extremity C of the arm ACC"A", the result is
BC=2sin q/2.
but from the isosceles triangle BAC we get
BC=2 BA sin 1/2 BAC,
wherefore, BA being equal to the unit of measure, the result is
q=BAC. (22)
SEC. II. We shall now show how this instrument is used in the two cases which interest us. To be specific we shall refer to the example: b=85°, c=45°, a=70°, whence a=72° 23'; and the arrangement of the diagram corresponds exactly to this example.
Given the three sides a, b, c, let the graduations b+c of the two principal arms coincide with the graduations b—c of the two auxiliary arms DE, FG; then revolve the two arms ACC"A" and OK until the index L coincides with the graduation a of the arm OK: the point C will then mark on the scale of the arm BH, the value of a.
A0=sin b+c/2 + sin b-c/2; AL=sin b+c/2 - sin b-c/2; OL=2sin a/2;
hence, in the rectilinear triangle AOL, from (21), we get BAC=a, and this angle will be exactly the one determined by C on the scale of BH. To obtain ? and y it will suffice to permute the data.
Given the two sides b, c, and the angle included a, let the two arms DE and FG be carried as in the preceding example to the extremity C to coincide with the point a of the scale BH, then revolve the arm OK until its graduated edge passes through the index L; this index, from what we have already seen, will determine on the scale of the arm OK the value of a. The angles ? and y can then be found as in the preceding case.
First Observation.—By considering the polar triangle, the instrument in question solves four of the six fundamental cases: then the doubtful cases are excluded; but the author of the pamphlet shows us that by starting with Napier's formula, it is sufficient, in order to solve even these two cases, to add to the instrument another scale, and to make use of an ordinary compass.
Second Observation.—The pamphlet we have been examining states that the degree of approximation obtainable with the trigonometric compass is five seconds (p. 9) ; that Borda conceived "a means of making the instrument just as exact as if its dimensions had been decupled " (p. 29) ; and the author of the same pamphlet adds, "the instrument interested me as much as if I had been the inventor of it" (p. 10).
The degree of approximation obtained in the numerical examples recorded is 10”; the dimensions of the instrument, however, are not mentioned anywhere in the article.
In any case we believe that this instrument deserves to be saved from oblivion. With the improvements made in the construction of geometrical instruments, it might, perhaps, become practically useful.