In a paper entitled "General Solutions of the Problems of 'The Reduction to the Prime Vertical and to the Meridian,' with a New Graphic Method for the Solution of Various Problems in Navigation," which appeared in the UNITED STATES NAVAL INSTITUTE PROCEEDINGS, Whole No. 151, May—June, 1914, the writer indicated how certain problems in nautical astronomy might be solved with slide rules.

These slide rules are shown on Plate II which accompanies this paper and with them all of the ordinary problems of astronomical navigation may be solved, including the "Time Sight" and the "Marcq Saint-Hilaire Method."

In the formulas used in explaining the use of the slide rules, which are given in the various problems, the letter in parentheses which follow any quantity indicates the scale on which that quantity is to be used.

RULES I AND V

These are the same, except Rule V has been made to a much larger scale and therefore its range is limited.

PROBLEM 1.—Find the azimuth of a celestial body, given its hour-angle, declination and true altitude.

sin *t (A) cos d (B)=* cos *h (C)* sin *Z (D).* (1)

Set *d *on scale *B *to* t* on scale *A*, then opposite *h* on scale *C* find the azimuth *Z *on scale *D.*

NAMING THE AZIMUTH

1. Mark the azimuth E. or W. according to whether the observed body is east or west of the meridian of the observer.

2. When *L=0°* give the azimuth the same name as the declination.

3. When the latitude and the declination are of the same name and the latitude is less than the declination, the azimuth has the same name.

4. When the latitude and the declination are of contrary names, the azimuth has the name of the declination.

5. When the latitude and the declination have the same name, and the declination is less than the latitude, the body will cross the prime vertical. When it is between the P. V. and the elevated pole, that is, when its hour-angle is greater than the hour-angle on the P. V., the azimuth will have the same name as the latitude. When the body is between the P. V. and the equator, that is, when its hour-angle is less than *t _{v}*, the hour-angle on the P. V., the azimuth will have the contrary name to the latitude.

The hour-angle on the P. V. may be found with Rule II as shown below.

The reasons for the above rules will be seen from an examination of Fig. 1, which shows the celestial sphere projected on the plane of the horizon.* M, M', M"* are the positions of three bodies.

By naming the azimuths in this way we avoid the use of angles greater than 90°.

PROBLEM 2.—Given the latitudes of the points of departure and destination *L _{1}* and

*L*, the G. C. D. between them

_{2}*D*, and their difference of longitude

*?,*find the initial and final G. C. courses

*C*and

_{1}*C*

_{2}.

sin *? (A) cos L _{2}(B)=cos(90°~D) (C)sin C_{1} (D).* (2)

Set *L _{2}* on scale

*B*to

*?*or (180°—

*?*) on scale

*A*; then opposite (

*90°~D*) on scale

*C,*find

*C*on scale

_{1}*D*. To find

*C*proceed as above, using

_{2}*L*in place of

_{1}*L*

_{2}.

NAMING THE COURSES

The vertex *V* of the G. C. track *AB *is the foot of a perpendicular let fall from the pole *P *on *AB* or *AB* produced, and is the point of highest latitude. When *V* falls between A and *B, C _{1}* and

*C*are each less than 90° (Fig. 2); when

_{2}*V*falls beyond

*A*or

*B*, or on

*AB*produced, as in Fig. 3, and

*L*, then

_{2}>L_{1}*C*<90° and

_{1}*C*90°, when reckoned from the elevated pole

_{2}>*P*; or in this case

*C*read from scale

_{2}*D*is given the contrary name to the latitude, or its supplement is given the same name. In all cases mark the course E. or W. according to the direction in which the ship is going.

NOTE.—In order to determine, when in doubt, whether *V* falls within or without the triangle *APB*, find *C _{1} *and

*C*with Rule IV. When both are less than 90°, reckoned from the same pole,

_{2}*V*is "in"; when one is greater and the other less than 90°,

*V*is "out," and nearer the point of greater latitude.

PROBLEM 6.—Given the hour-angle of a body on the prime vertical, its declination and the latitude of the observer, find its altitude on the prime vertical.

NOTE ON RULES I AND V.—Scale *A *should have been laid off in hours and minutes. Due to oversight the scale of degrees was inserted and it was necessary to place the scale of hours and minutes above this, as no time was available to make new drawings. As Rules I and V now stand scales *C *and* D* are not necessary, but they are useful in that they simplify the instructions for using the Rules.

RULE II

Any formula deduced by Napier's First Rule may be solved with Rule II, thus in all cases problems involving right spherical triangles may be solved with either Rule I or Rule II.

PROBLEM 7.—Find the hour-angle of a celestial body when on the prime vertical, given its declination and the latitude of the observer.

PROBLEM 8.—Given the latitudes of departure and destination, and the latitude of the vertex, find the longitudes of the vertex from the points of departure and destination.

PROBLEM 9.—Given the latitude of the vertex and the longitudes from the vertex to the points of departure and destination, find the distances from the vertex to these points and the total distance along the G. C. track.

PROBLEM II.—To lay down a great circle track on a Mercator chart.

1. Find the initial and final courses with Rule IV, Problem 15.

2. These determine whether the vertex is "in" or "out."

3. Find the latitude of the vertex with Rules I or V, Problem 4.

4. Find the longitudes of the vertex from *A* and *B*, *? _{A}* and

*?*, Rule II.

_{B}

5. Plot the position of the vertex if "in."

6. Determine a sufficient number of points in the track with Rule II and draw a curve through them with a flexible batten or connect them by straight lines.

Points may be determined in the track by assuming latitudes ranging between those of departure and destination and that of the vertex, and finding the corresponding longitudes. Thus, if *L _{p}*, be the latitude of any point in the G. C. track and

*?*be its corresponding longitude from the vertex, then for each value of

_{p}*L*two points in the track are determined by -.±

_{p}*?*, as each parallel of latitude intersects the G. C. track in points whose longitudes from the vertex are equal. From (8), we have

_{p}

Set *L _{v }*on scale

*F*to the assumed latitudes, taken in succession, on scale

*E*and find the corresponding longitudes on scale

*G*opposite the mark

*X*on scale

*H*.

If preferred the longitudes may be assumed and the corresponding latitudes found thus:

Set the longitudes in succession on scale *G* to the mark *X* on scale *H*, and find the corresponding latitudes on scale *E,* opposite *L _{v }*on scale

*F*.

In order to shape a G. C. C. from time to time during a voyage, note that if the initial G. C. D. be known, or is computed, it will not be necessary to recompute it during the voyage, as a rule, as the distance to the destination may be found every time it is desired to change the course by deducting the total distance made good from the initial distance and thus obtain the data for using Rules I or V for obtaining a new G. C. C. as often as may be desired.

RULE III

PROBLEM 12.—Given the rate of change of altitude per minute *R _{m}* of a celestial body and the latitude of the observer, find the azimuth of the body.

Set 1 on scale *L* to *R _{m} *on scale

*M*and find

*Z*on scale

*J*opposite the latitude

*L*on scale

*K*.

*Example 17*.—At sea *L*=12° N. From the difference between two observed altitudes of an unknown star and the interval between the sights its rate of change of altitude per minute was 13.3'. The star being in the S. E. quadrant, what was the true bearing of the star?

Ans. S. 64° 45' E.

THE REDUCTION TO THE MERIDIAN BY THE MEAN AZIMUTH METHOD

This method was fully demonstrated by the writer in the UNITED STATES NAVAL INSTITUTE PROCEEDINGS, Whole No. 151, May— June, 1914.

PROBLEM 13.—Given the latitude by D. R., the true altitude, hug-angle and bearing or azimuth of a known celestial body, find the "reduction *?h*" of this altitude to the meridian altitude and thence the latitude of the observer.

Set the latitude by D. R., *L*, to ½*Z* on scale *J* and find the reduction to the meridian *?h* in minutes of arc, on scale *M* opposite* t*, expressed in minutes and decimals of time on scale *L.*

PROBLEM 17.-Given the latitude and longitude of the observer, the hour-angle and altitude of an unknown star, and the time of observation to find its right ascension and declination, with Rule I.

(1) When the star is east of the observer's meridian add its hour-angle to the L. S. T. to obtain the R. A. of the star; when west subtract it from the L. S. T.

(2) Set *h* on scale *C* to *Z* on scale *D* and find *d* on scale *B* opposite *t* on scale *A.* (Use Rule I or V.)

NAMING THE DECLINATION

There will be no difficulty in determining the name of the declination except when the declination is small, or when the star is near the equator. In all cases the sign of the declination may be readily determined by Table I, for which the writer is indebted to Commander H. L. Rice, U. S. Navy.

DIRECTIONS FOR CONSTRUCTING THE SLIDE RULES

These should be mounted so that Rules I to V form one set and Rules VI and VII the other.

Paste the first set on good stiff cardboard, using good mucilage and not photo paste. Spread the mucilage on the cardboard with a good brush about 2" wide, and place the print over it and rub down quickly and evenly, first very lightly lengthwise, then crosswise so as not to stretch the paper. Use a blotter or piece of paper between the print and cloth. Weight down the cardboard and let it dry.

With a safety-razor blade and a steel straight-edge or a carpenter's square to guide it, cut the inner scales *B, C, F, G,* etc., from the mounted print. Or if convenient have the mounted print cut into strips with a stationer's paper-cutting machine.

Provide sufficient cardboard of the same thickness and about 26" wide, to have a base 8" x 26", on which to mount the scales, and to make covering pieces and filling pieces.

Assemble these various parts, as shown in Fig. 5, by laying the filling piece 2 in the position shown and make a mark along its inner edge with a sharp pencil. Spread mucilage evenly on the base 1, leaving a margin of ?" from line *p*. Put mucilage also on piece 2, leaving ?” margin from its inner edge. Place 2 on 1 and weight it down to set.

Next spread mucilage on top of. 2 and on bottom of 3, leaving ?" margin from *n*. Place 3 on 2 and weight down to set. Next slip the slide 4 under 3 and move it back and forth to see that there is no excess of mucilage. Press 4 gently up against 2 and make a pencil line along the edges *m* and *q*. Remove 4 and spread mucilage evenly on the bottom of 5 and top of 4, leaving ?" margins at *m* and *q*. Place 5 carefully on 4, rub down, and place the slide in the position shown in the figure. Move it back and forth and see that the edges *A* and *B* coincide their full length. Mark the lines *r, s, t* and *u*. Put 6 in place and move the slide back and forth. Then place 7 in position and press it gently against 5 until the edges *C *and *D* coincide throughout. Move the slide back and forth until it moves freely.

Weight with books and 'set aside to dry.

In the same way assemble Rules II, III, IV and V.

It is important that the outside scales *A *and *D* should be lined up exactly as shown on the print. This may be easily done by setting the slide to scale *A *as it was originally and lining scale *D *up by it. After the rules have been assembled on the base it should be glued, or secured with thumb tacks to a piece of ¾" board.

While it is intended to have these slide rules printed on cardboard, all of the examples in this paper under Rules I, II, III, IV and V were solved with a set of rules made as shown above. The results compare favorably with those obtained by computation and are sufficiently accurate for practical navigation.

To obtain accurate results with Rules VI and VII, however, these should be printed on cardboard, as a slight uneven distortion of the paper when working to seconds of time will of course, materially affect the result.

If the rules be assembled as directed above, the navigator can, with the expenditure of a small amount of labor and care, make himself a set of slide rules which, for all practical purposes, will be just as good as though they had been made by an instrument maker at a cost that would in all probability put them far beyond his reach.

It would probably be better to make Rules VI and VII on discs so as to have the scales much longer without making the rules unwieldly. The writer hopes to accomplish this at some future time.