H. K. F. Courses. Winds. Leeway. REMARKS. Wednesday, September 3, 1823. 7 A.M. E N NE 7 Strong gales and squally weather. Hauled the main sail up, and handed it, and set the main staysail. Heavy gales and rain. Hauled the foresail up, set the Hauled down the fore and main staysail, got the jibboom 4 5. 6 7 Up W 8 Off SWbS 7 7 Ditto weather. Ship under reefed trysail. Gale increasing, with a heavy sea. Handed the main topsail. Heavy sea. Took in' trysail, and lay to under bare poles. Allow 1 mile per hour for drift while lying to. Variation 2 Points W. It may be observed, in the first place, on the preceding days' works, that as the variation is westerly, it must be allowed to the left of the compass courses ; and therefore when the ship makes leeway on the larboard tack, the difference between the leeway and variation is the correction to be applied to the course, to the left if the variation is greater, but to the right if the leeway is greater. When the ship makes leeway on the starboard tack, the allowance for it, as well as that for variation being to the left, their sum will be the correction to be applied to the compass course ; and when no leeway is made, the only correction is for the variation. Now EN E, the opposite point to the bearing of the land, is the first course, and this and the drift being corrected for variation, and the other courses for variation and leeway, (when there is any) and the distances on each added up, we have the following Lat left 37° 48'N Mer pts 2453" Lon obs 25° 13' W 39 Lon acct 24 11 See at p. 190 the method of finding the diff long 1.6 by inspection. 2.8 From 11 A. M. till noon the ship's true course is 2:4 NW, nearly 5 miles, whence the diff lat is also .6 nearly 5 miles; which, added to the latitude by .5 double altitudes at 11 A. M. gives 38° 20' N for the 1.0 latitude by observation at noon. With this latitude 8:9 and the long by account, the bearing and distance of the Lizard are found to be N 493° E, 1075 miles W. 8:0 The courses for the second day's work being first corrected for leeway only, as the variation is given in degrees, we have the following In the third day's work, from 3 A. M. the ship in the heavy gale is laid to, 1 mile an hour being allowed for her drift. The courses being corrected for leeway and variation, we have the following In the above days' works the knots are divided into 10 parts for fathoms, and the distances are taken in the Traverse Tables to fathoms. This however is a degree of exactness which in general practice is quite unnecessary, as the courses and distances can seldom be determined with sufficient exactness to justify any confidence that the results deduced from them at the end of a dlay are within less than two or three miles of the truth. D S. ON THE QUADRANT, AND OTHER REFLECTING INSTRU MENTS FOR MEASURING ANGLES. The quadrant, sextant, and reflecting circle, are in principle the same instrument, and the following is an explanation of the principle on which they are constructed. Let A B C be a section of a reflecting surface, F B a ray of light falling upon it, and reflected A с again in the direction B E, and B D a perpendicular at the point of impact; then it is a well-known optical fact, that the angles F B C and E B A are equal, and that F B, D B, and E B are in the same plane. Again, if AC were a reflecting surface, and a ray of light, S B, from any celestial object S, were reflected to an eye at E, the image of the object would appear at S' on the other side of the plane, the angles SB A and A B S', as well as E BC, being equal ; and if E B bear no sensible proportion to the distance of S, the angles S E S and SBS may be considered as equal; for their difference, BS E, will be of no sensible magnitude. These principles being premised, let the quadrant or sextant be represented by BHK, a sector of a circle, having a revolving radius B I, which carries with it round the centre, a mirror A B, whose plane is perpendicular to that of the sector: and, for the LE. sake of simplicity, let us sup M pose that B I is in the plane of the mirror. Let CD be F another mirror which is parallel to A B when B I coincides with BK; and let AB and CD, the planes of the K mirrors, meet when produced in G. Let S be a celestial object, and S B, B D, DE L', the course of a ray of light proceeding from it, and reflected from the mirrors AB and C D in succession to an eye at E, where D E meets, S B produced, and conceive B D to be produced to F. Then at E the object will appear at S, and its image at D, and the angle BED will be the apparent distance of the object and its reflected image; and whether the eye B В D H I and its reflected image at D, will be the same, if the distance of the eye from E be evanescent with respect to the distance of S. Now the angle SB A being equal to the angle D BG, by the above-mentioned property of reflection, and equal also to the vertical angle E B G, the angle D B E is bisected by B G; and B D C being for a like reason equal to each of the angles FD G and EDG, the angle EDF is bisected by DG, hence (Geom. Theo. 23.) the angle B GD, or the alternate angle G B K, which is measured by I K, is equal to half the angle B E D, the angular distance of the object from its reflected image. In the quadrant, sextant, &c. the half degrees on the arc H K are considered as whole degrees, and therefore the angle read from that arc is the measure of the distance of the object from its reflected image. If MN were a mirror perpendicular to CD, the angles BDN, E' D M would be equal, and therefore the ray B D would be reflected from M N in the direction D E' L, opposite to its direction as reflected from C D. Hence, as seen by an eye in the line D E' L, the arc I K will be the measure of the supplement of the distance of the object from its reflected image. When the distance of an object from its image is determined from the inclination of two mirrors situated as A B and C D are, the eye in the line D E L' looking towards the object, the angle is said to be measured by a fore observation; but when the supplement of the distance of the object from its image is obtained from the inclination of two mirrors situated as AB and M N are, the eye in the line DE' L looking from the object, the angle is said to be determined by a back observation. The mirror A B is called the index glass, CD the fore horizon glass, and M N the back horizon glass. The horizon glasses are only partly silvered that objects may be seen through them, as well as by reflection from their surfaces. When the image of an object, as S, reflected from the index and horizon glasses to the eye at E, is in apparent contact with an object, as L, seen by the eye through the horizon glass at the same instant, the arc I K measures the angular distance of the objects S and L. The revolving radius B I is called the index, and it is evidently of no importance, provided they revolve together, that it and the index glass A B should be in the same plane. The arc H K, called the limb, is generally in quadrants graduated to 20', and the index carries a scale called, from its inventor, a Vernier, the length of the divisions on which are 19'; hence the division on the vernier, which corresponds with a division on the limb, points out the minutes which the beginning of the vernier scale has advanced beyond the preceding division on the limb. In sextants and circles, the divisions on the limb and the vernier are generally more minute, but the principle of division is the same, the following being the general theory of it. If n be the minutes in each division on the limb, and m = the number of those divisions which are taken as the whole length of the vernier scale, then m n is the minutes of the limb which the vernier comprehends; and this is divided into m + 1 equal parts for the length of each division on the vernier. Hencen m ti m n n 19', n n = the difference between the lengths of the vernier and m+1 limb divisions, or it is the length of the parts into which the vernier scale subdivides the divisions on the limb. If n = 20', and m = then 1', if n = 10' and m = 59', then = 10', &c. m + 1 m +1 If when A B and C D are parallel, (see the last figure,) or A B and M N perpendicular to each other, the beginning of the divisions on the vernier does not coincide with the beginning of the divisions on the limb, their distance is called the index error, subtractive, of course, from the arc read from the limb, as the measure of an angle, when the vernier division is to the left, and additive when to the right of the first division on the limb. If the first divisions on the scales coincide when the mirrors are parallel, the instrument has no index error, and the arc pointed out by the index is the measure of the required angle. As the angle S EL is double the angle B GD, the image of S will coincide with itself, when A B and C D are parallel ; and, for a like reason, when A B and M N are perpendicular, S will be diametrically opposite to its image as reflected from MN. A telescope is sometimes applied to guide the sight, and, from its magnifying power, to enable the observer to mark the contact of the objects observed with greater exactness. The adjustments of the instruments are, to make all the mirrors perpendicular, and the axis of the telescope parallel to the plane of the instrument; the fore horizon glass parallel, and the back one perpendicular to the index glass, when 0 on the vernier coincides with 0 on the limb. There are various screws for making these adjustments, the the method of doing which, as well as of using the instruments, will be best learned by practice, under the direction of a skilful teacher. The reflecting circle differs from the quadrant or sextant chiefly in having the complete circle graduated, and the measure of the required angle pointed out by three indexes placed at equal distances from each other, one of which, having a screw attached to it, is called the leading index. Angles are measured by this instrument on each side of 0 on the arc, by reversing the face of it, and the mean of the measures on each side of o is the true measure of the angle, independent |