The ingenious method, erroneously called "The New Navigation," which was invented thirty-four years ago by Admiral Marcq Saint Hilaire, French Navy, for determining lines of position at sea, is generally recognized nowadays by intelligent navigators to be superior to all others. This method is also superior to all others for determining geographical position at sea when the position given by account (D. R.) is in doubt. Although better results are obtained by determining longitude when the latitude is known *exactly* (which often happens during the twilight and on moonlight nights by the observation of stars and planets), or by determining the latitude when the longitude is known *exactly*, yet we prefer to treat *all *observations by Marcq Saint Hilaire's method. In the particular cases mentioned, the true position of the ship can by means of a small correction be easily deduced from the position found by the method of Marcq Saint Hilaire from the position by account (D. R.).

If *A* is the ship's position by account and *BB'* the line of position, *B* will be the ship's position by Marcq Saint Hilaire's method, *C* the ship's position by determining the longitude by means of the latitude, and *D* the ship's position by determining the latitude by means of the longitude (Fig. 1).

The distance between *A *and *B* is the difference between the true altitude and the calculated altitude (commonly called the *zenith difference*) and the angle S1AG is equal to the body's azimuth. Therefore, when *AB* and angle *S1AG* are known, it is easy to determine the corrections.

The advantages of the method of Marcq Saint Hilaire are due to its exactness and the generality of its application and, as Commander W. C. P. Muir, U. S. Navy, Head of the Department of Navigation, U. S. Naval Academy, explicitly states in heavy type in his excellent treatise on "Navigation and Compass Deviations," 1906, page 638 : *"it is available practically without limitations as to azimuth, altitude, or hour angle, and may be used to work a sight whether a time sight, a Ø” Ø’ sight, or one of a body observed near the meridian"¹*

¹Vide for details the author's "O Methodo de Marcq Saint Hilaire," Imprensa Nacional, Rio de Janeiro, 1902. Reprinted from the *Revista Maritima, Brazileira, *Nov. 1899, Jan. 1900, and Oct. 1902, by order of the Minister of Marine.

There is but one class of exceptions to the proceedings by which we lay down the line of position as a straight line according to Marcq Saint Hilaire's method and that is where all other methods fail too: it is when the altitude of the observed body is greater than 85°. In this class of cases the straight line can no longer be considered a practical substitute for the circle of position on account of the smallness of the zenith distance, but in these cases the actual position line or circle of position may be readily laid down upon a chart or elsewhere.²

The determination of geographical position and of a line of position by Marcq Saint Hilaire's method requires a knowledge of the ship's position by account (D. R.) from which the altitude and azimuth of the observed body due to this position of the observer are calculated.

In Fig. 2., *P* is the elevated pole, *Z *the zenith by account, and *M* the observed body whose altitude is *h* and declination 8. *PZ* is the colatitude = 90°— *L*.

If we let fall a perpendicular from *M *on *PZ*, it will divide the triangle of position into two right-angled triangles, and altitude and azimuth can be easily determined by solving these two triangles.

²Vide an article by the author : "Limites de coincidencia da recta Marcq Saint Hilaire com a curva de posicão correspondente” in the *Revista Maritima Brazileira,* July, 1906, page 41. For an altitude of 86° the useful part of the straight line of position is 44 miles long. For 89° it is only 22 miles. For a given latitude the most favorable circumstances for determining lines of position are when the observed body's hour angle, declination, and zenith distance are the greatest practicable.

³Vide *Chauvenet,* A Treatise on Plane and Spherical Trigonometry, 9th ed., Philadelphia, page 169, or the author's " Estudo elementar de Trigonometria Espherica e algumas das suas applições a Astronomia Espherica, Navegação e Geographia," edited by H. Garnier, Paris and Rio de Janeiro. 1903, page 26.

From well-known theorems in Spherical Trigonometry³ we obtain the following two groups of formulæ which bind together several of the elements of the two right-angled triangles:

These two independent groups of four formulæ each, appearing now for the first time locked together on account of their similarity, permit the simple, easy, and rapid determination of altitude and azimuth and of other elements in various other problems which can be solved by means of these equations.

In order to facilitate these determinations many tables based on the division of the triangle of position into two right-angled triangles have been devised, constructed and published, and the perpendicular has either been dropped from the body on *PZ *or from* Z* on *PM*.

Lord Kelvin's hour angle and azimuth tables,? which are more naturally adapted for calculating altitudes and azimuths, as we have shown in our improved edition of his tables,? are based on the 3d and 4th formulæ of each of the two groups, while Fuss' altitude and azimuth tables^{6} are based on the 1st and 4th formulæ of each of the two groups.

A general review of these tables and many others, and also of Favé and Rollet de 1'Isle's^{7} and Littlehales'^{8} graphical tables has been made by Dr. Giuseppe Pesci of the Royal Naval Academy of Livorno, Italy, in a very interesting study entitled, "Resolução Nomographica do Triangulo de Posição," translated from the Italian into Portuguese by the author of this article and recently published in the three numbers of the *Revista Maritima Brazileira* for November and December, 1907, and February, 1908.

^{4}Tables for facilitating Sumner's Method at Sea, London, 1876.

^{5}A Navegação sem logarithmos, Imprensa Nacional, Rio de Janeiro, 1903.

^{6}Tablizti dlya Nakasdeniya Visott i Azimutoff, Saint Petersburg, 1901.

^{7}Abaque pour la détermination du point à la mer, *Annales hydrographiques, *Paris, 1892. *Revue maritime et coloniale*, Paris, January, 1893.

^{8}Altitude, azimuth and geographical position, Philadelphia, 1906.

They all have their advantages and disadvantages, but the disadvantages are especially inherent in the numerical tables on account of the required numerical interpolations which lead to errors and loss of time, not to mention the turning of pages to find the data.

For these reasons, I have looked for *Nomography*^{9} to give an easy and very quick solution of the problem and I hope this article will be of interest to astronomers and navigators and also to naval tacticians and ordnance officers, and serve to stimulate them to work out many of their problems by *Nomography*. I hope also that Dr. Pesci's nomogram for solving the equations of the 2d group of formulæ will prove useful as a substitute for the time-azimuth tables actually in use, since these present the same inconveniences as all tables, and that both nomograms will prove useful for check work which will be greatly facilitated and expedited by their use.

To Dr. Pesci I owe my enthusiasm for *Nomography,* and the recognition of its important possibilities for solving easily and rapidly various problems in navigation, naval tactics, balistics, target practice, etc.

LAFAY'S NOMOGRAM.

All four of the formulæ of the first group are of the same type, namely,

sin *a* = sin*? *sin *y*

for which Captain Lafay constructed, in 1895, a very interesting nomogram^{10} in connection with his research work in elliptic polarization. This nomogram is based on d'Ocagne's ingenious method of alined numbered points.

^{9}Vide *Maurice d'Ocagne,* Traité de Nomographie, Paris, 1899 and Exposé synthétique des Principes fondamentaux de la Nomographie, 1903. (Extrait du *Journal de l'Ecole Polytechnique*, 1903); *Schilling*, Ueber die Nomographie von M. d'Ocagne, Leipzig, 1900; *M. J. Eichhorn*, The construction and use of graphical tables (*Western Electrician, *Chicago, March 9, 1901, page 162); *Soreau*, Contribution à la Théorie et aux Applications de la Nomographie. Mémoirs de la Société des Ingenieurs Civils de France, August, 1901. *Ricci*, La Nomografia, Rome, 1901, G. *Pesci*, Cenni di Nomografia, Livorno, 1901. Sobre Nomografia Elemental. Revista Trimestral de Matemáticas, Zaragoza, año V, 1905, pages 138-161, and Lezioni di Nomografia, Livorno, 1905.* Perret*, Annales hydrographiques, Paris, 1904.

^{10}Journal de Physique théorique et appliqué, 3eme serie, t. IV, April, 1905, page 178.

A very elementary demonstration of it is given by *Soreau* (loc. cit.) as follows:

Let us take for the axes of co-ordinates three lines meeting in the same point in such a manner that one bisects the angle made by the other two.

It can be easily shown that if three points, *A, B,* and *C,* whose respective co-ordinates are *x, y*, and *z*, are in a straight line or *alined, *they satisfy the following equation

__I__ + __I__ = __I__

*x y z*

which appears in Optics, as a relation between focal distances. Now

__OD__ + __CD__ = I.

* OA OB *

In the isosceles triangle *COD*

* OD = CD = OC*

* 2 cos ^{?} *

And therefore

__I__ + __I __ = __2 cos ^{?}__

*OA OB OC*

If we take

* _{X }= OA = l.f_{1}*

* _{Y} = OB = l.f_{2}*

* _{Z} = OC = 2lcos?.ƒ_{3}*

We will have __I__ + __I__ = __I__

ƒ ƒ_{2} ƒ_{3}

where ƒ_{1}, ƒ_{2}, and ƒ_{3} are three functions and* l* a certain factor of amplification or reduction.

By a very ingenious artifice, Captain Lafay shows how the equation

ƒ_{1 }+ ƒ_{2 }= ƒ_{3}

(or ƒ_{1 }+ ƒ_{2 }= ƒ_{3} after applying logarithms to both members) can be transformed in an infinity of different ways of multiplying each function by *k *and adding *2h* to each member of the equation.

Thus

(*h + kƒ _{1}*) + (

*h + kƒ*) =

_{2}*2h + kƒ*

_{3}or

__I + I = I__

__I I I__

* h + kƒ _{1 }h + kƒ_{2} 2h + kƒ_{3}*

This transformation is necessary and advantageous when the functions vary between o and ± ∞, thus giving a finite dimension to the scales, which otherwise would be infinitely long.

If, according to Captain Lafay, we take *h* = 1, *k* = 10, and * _{?}* = 30°, supposing

*l*= 72

^{mm}, the nomogram will take the form of Fig. 4.

It is only necessary to calculate the values of *y*. The values of *z *are readily found by alining the point ? = 0° with the successive numbered points on *t* (values of *y*). The intersections of these alinements with the bisector will give the numbered points corresponding to the different values of *a*, hence the value of *z*. These numbers are the same as those corresponding to *y*, because the equation shows that when ? = 0°, sin *a* = sin *t *or *a* = *t*. The positions of these points can be checked by using the point *t* = 90° and alining it with the points corresponding to the successive values of ? (values of *x)*. The numbers in this case are the complements, because when *t* - 90°, sin *a* = cos ? or* a* = 90° — ?

When two of the quantities are given in the equation sin *a* = cos ? sin *t* to find the third, it is only necessary to aline the numbered points corresponding to the given data and read the numbered point where the line intersects the proper scale. A visual interpolation is generally necessary for points between the numbered ones. This is done in the same way as on any scale.

*Examples:* I.? = 30°, *t* = 35°, we find *a* = 29°, 8. II.* a* = 40°, ? = 33°, we find *t* = 50°. III. ? = 58°, *a *= 30°, we find *t* = 70°.

DETERMINATION OF *B.*

When the perpendicular falls within the triangle of position as in Fig. 2, we have

*PZ *= 90° — *L *= 90° — *b *+ 90° — *B*

and therefore

*B *= 90° + *L — b*

If each one of the other three distinctive positions a celestial body can occupy, according to its declination and hour angle and the latitude of the observer, were considered, we would have the following precepts for finding *B*:

A complete discussion of these formulae was made recently by Dr. Pesci in the *Revista Maritima Brazileira *(loc. cit.) and therefore we will not repeat it here.

By these formulae *B* can be obtained from* b* and *L* without giving consideration to algebraic signs. The brackets show that the quantities in them ought to be added first in order to simplify calculations.

The determination of *B* is thus very simple. In the first two cases, which are the most frequent in practice, 90° is always added to the smaller of the two quantities *b* and *L*; and from this sum is subtracted the larger of the two.

In the third and fourth cases *b* and *L* are always added together. If their sum is greater than 90°, 90° is subtracted from it. If the sum is smaller than 90°, it is subtracted from 90°.

The precepts and formulae are set forth on the large scale nomograms as a convenient aid to the memory in practical work.

We will now show how with Lafay's nomogram and the formulae of the first group, altitude and azimuth can be easily and rapidly determined, given the hour angle *t*, the declination of the body ** (insert symbol here), **and the latitude of the observer

*L*.

The key to the nomogram is as follows:

Example:* L* = 20°S, ? = 30°S,* t* = 35°. We will find *a* = 29°, 47’, *b *= 35° 11’, *B *= 74° 49’ and *h *= 56° 53’ and* Z* = 65°, 4. In this case the azimuth is called a time-altitude-azimuth.

DR. PESCI'S NOMOGRAM.

All four of the formulae of the second group are of the same type, namely,

tan a tan *? = *sin y

for which Dr. Pesci has just published a nomogram (*Revista Maritima Brazileira* for February 1908, page 1036).

It will be readily seen that all the elements in Dr. Pesci's formula are the complements of those in the formulae of the second group.

This nomogram, as well as Lafay's, is based on d'Ocagne's method of alined numbered points; and an elementary demonstration of it is given by Dr. Pesci on page 1037 of the *Revista Maritima Brazileira* for February, 1908.

Fig. 5 shows the nomogram for cot *b* cot (90° — ** insert symbol here**) = cos

*t*with

*l*= 100

^{mm}. The scale on the right, parallel to the one on the left, is an ordinary scale of natural cosines on which are read

*t, Z, b,*and

*B*in the second members of the second group of formulae. The graduation of the diagonal scale is made by the following relation:

where* l* is the length of the parallel scales and also the distance between them. The scale on the left is an ordinary scale of cotangents, and, as will be noticed, runs only from 45° to 90°. This valuable restriction placed on the scale permits a greater approximation to be obtained from it, and is a consequence of the nature of the equation

cot *b* cot (90° — ?) = cos *t*.

This equation cannot be satisfied when *b* and 90° — ? are at the same time smaller than 45°, because then cos *t* would be greater than *I*, which is absurd. Therefore, when values of *b* and 90° — ? are given, the one that is larger than 45° is always taken on the left-hand scale and the other smaller than 45° on the diagonal scale. If this procedure is not followed the alinement will be impossible. When both are larger than 45°, it will be indifferent on which of the two scales they are read. However, in this case, for geometrical reasons, it is better to read* b* or 90° — ? on the left hand scale and to take 90° — ? or *b* from the diagonal scale.

This nomogram, on account of its peculiar shape, is called a "nomogram in Z." It might be more properly called a "nomogram in N," because it can be better worked upright.

In the equation cot *b* cot (90° — ?) = cos *t*, to find any one of the three quantities when the other two are given, it is only necessary to aline the numbered points corresponding to the given data and read the numbered point where the line intersects the proper scale. A visual interpolation is generally necessary for points between the numbered ones. This is done in the same way as on any scale.

Examples: I. *b* = 69°, 90° — ? = 56°, we find *t* = 75° II. 90° — ? = 62°, *t* = 38°, we find b = 34°. III. *b *= 74°,* t* = 54°, we find 90° —? = 26°.

The key to this nomogram when used for determining altitude and azimuth is as follows:

*b *is determined by 90° — ? and *t.*

90°— *a* is determined by *t* and *b*.

*Z* is determined by 90° — *a* and *B*.

*h *is determined by *B *and *Z*.

With these formulae and the nomogram, it is not necessary to read 90°— *a* on the scale because *Z* is determined by 90° — *a* and *B*. This is important and gives *h* and *Z* independently of *a*, while all the elements are determined by their cotangents, and hence are the best results possible.

This is why Dr. Pesci's nomogram gives better results than Lafay's in the determination of altitude and azimuth and other problems.

*Example: L* = 42° N, ? = 23° 5’ N, *t* = 49° 41’. We will find *b* = 33° 23’, *B* = 81° 23’, 90° — *a* = 45° 27’, and *Z *= 180° — 81° 20’ and *h* = 44° 48’. In this case the azimuth is called a time-azimuth.

If *t *= 90°,* b* is also 90° and then 90° — *a* cannot be determined by the third formula of the second group. But the first and third formulæ of the first group show ? = 90° — *a* when* t* = 90° and *Z* can be determined.

cot *Z* cot ? = cos *B*

If* B* = 90°, *Z *is also 90°, and then *h* cannot be determined by the second formula of the second goup. But the second and fourth formulæ of the first group show that *h* is then equal to 90° — *a*.

TIME-AZIMUTHS.

Time-azimuths can be easily and rapidly determined according to the first, third, and fourth formulae of the second group, by Dr. Pesci's nomogram.

Horizon-azimuths depending upon the following formula

Sin *Z* = cos ? sin *t*

when ? and *t *are known, and

__sin ?__

cos *Z* = cos *L* or sin ? = cos *Z *sin (90° — *L*)

when ? and *L* are known, can be determined by Lafay’s nomogram. In this case the hour angle is given by

cos *t* = — tan *L *tan ?

and is easily determined by Dr. Pesci’s nomogram.

IDENTIFICATION OF CELESTIAL BODIES.

The problem of identifying celestial bodies is the reverse of the problem of determining altitude and azimuth, and, therefore, we ought to begin with the formula in *h* and *Z* and proceed backwards until ? and *t* are found.

COURSE AND DISTANCE FINDING.

The problem of finding course and distance in Great Circle Sailing is the same as the problem of determining altitude and azimuth. The zenith distance or complement of the altitude corresponds to the distance, and the azimuth to the course. The only difference is that the distance between the two given points can be greater than 90°, whereas the zenith distance cannot be greater than 90°.

Our purpose is not to work out the details of each one of these problems, which are well known to all astronomers and navigators; but rather to show the advantages of solving them by means of the two nomograms, and especially by Dr. Pesci's.

It is now well known that nomograms, with their graphical readings, do away with the complications of arithmetical interpolation, replacing it with a visual interpolation and thus minimizing the chances of error and avoiding the necessity of turning the pages of tables.

As the nomograms correspond to an infinity of graphical constructions, it is desired to point out that no graphical constructions are necessary, but only graphical readings in order to solve the equation to which they belong.

In order to facilitate the alinement of any three points Messrs. Keuffel and Esser of New York were employed by me to make of xylonite a transparent straight-edge of 28½ inches in length, with a straight black line drawn through the middle of it. This is a more advantageous form of ruler than those heretofore proposed, since its use greatly facilitates the rapid alinement of the three points and permits an accurate and refined graphical reading. This ruler could also be made of glass.

Naturally the approximation obtainable from a given nomogram depends on the length of its scales, and on the accuracy with which the readings are made. For this purpose good eyesight is very valuable. Greater approximation may be obtained by the aid of a magnifier as used by engravers or with slide rules.

Large scale nomograms have been engraved upon copper plates. They are carefully printed on paper and published by the U. S. Naval Institute, and may be obtained upon application to the Secretary and Treasurer.

In Dr. Pesci's nomogram the scale is 500^{mm} long occupying an area 500^{mm} x 500^{mm}. With it *h* and *Z *can be obtained with an approximation of 1' (and in many cases with less than 1') which is sufficient for practical purposes.

This nomogram with a scale equal to 250^{mm} giving *h* and *Z *with an approximation of *2'* (and in many cases with less than *2'*), is large enough for time-azimuths, star-identification, and check work. The nomogram is then much more handy.

In Captain Lafay's nomogram the outside scales are 360^{mm} long and the approximation obtainable varies with the parts of the scales employed in solving the problem. However, for practicable purposes we are of the opinion that the scale of 360^{mm} is sufficiently large.

We have limited the size of our nomograms in view of their use on board ship. For office work, where space and handiness are not so important a factor as on board ship, their size can be increased if greater approximation is desirable.

With this end in view they might also be constructed on such substances as celluloid, boxwood, aluminum and ivory; and the scales be engine-divided.

In conclusion, I wish to express to Mr. Geo. W. Littlehales, Hydrographic Engineer of the U. S. Navy Department, my hearty thanks for his kindness in revising this article and for many valuable suggestions regarding the construction of the nomograms.

WASHINGTON, *March 24, 1908.*