The following text may have been generated by Optical Character Recognition, with varying degrees of accuracy. Reader beware!

32 Any two sides of a triangle added together are greater than the third side, . 'The greatest side (of course, distance or hypotenuse) of a triangle is opposite the greatest angle, and the last side op- posite the least angle; also, in the same triangle, equal sides are opposite to equal angles. remember this since the angles will help you with sides and vice versa. In every right-angled triangle there must be two parts given, besides the right angle (which is always known), to find the other three; and one of these parts must be at least a side, because the angles will only give the proportion, not the absolute length of the sides. In every- such triangle one of the sides must be considered as represent- ing the radius of a circle; then, 1. If the hypotenuse be made radius, the perpéndicular and base will be the Size of its opposite angle. 2. If the base be made radius, the perpendicular will be the tangent of its opposite angle, and the hypotenuse the secant of the same angle. r A 2 9 TANGENT The sine of an arc is a line drawn from one end of the arc perpendicular to a diameter drawn through the other end of the same arc; thus, RS is the sine of the arc AS, RS being a line drawn from one end, S, of that arc perpendicular to DA, which is the diameter passing through the other end, A, of the arc. The tangent of an arc is a right line drawn ' perpendicular to the diameter, passing through one end of the arc and terminated by a line drawn from the center through the other end hoe arc; thus, AT is the tangent of the arc The secant of an arc is a right line drawn from the center through one end of the arc to meet the tangent drawn from the other end; thus, CT is the secant of the arc AS. 3. If the perpendicular is made ra- dius, the base will be the tangent of the opposite angle, and the hypotenuse the secant of the same angle; the sine, tangent, or secant of an angle, in each case, to co-sine (complement), co-tan- gent, or co-secant of the other. Thus, if in the triangles ABC, the hypote- nuse AC be considered as radius of a circle, then it is evident the perpendicular BC will be It is important to: JAE Marine Review the sine of the angle A, and the base AB the sine of the angle C; if 'the base AB be made radius, then the perpendicular BC will be the tangent of the angle A, and the hypotenuse AC the secant of the angle A; if the perpen- dicular BC be made radius, then the AB will be the tangent of the angle C, and the hypotenuse the secant of angle C. To find a side: Consider any one of the sides as representing the radius of "a circle, and write upon it the word radius; then upon the other sides write the parts they represent according to the preceding rules, which call the names of the sides; then say: As the name of the given side Is to the given side, - So is the name of the side fequied To the side required. To find an angle: Consider one of the given sides as' representing the radius of a circle, and write upon it the word radius; and upon the .other sides write the parts they represent ac- cording to the preceding rules; then say: 3 "As the side beet tpating radius -Is to radius, So is the other. given side To the sine, tangent, or secant of the angle by it represented.) In order to exemplify the foregoing rules we will take the example have given: the base and perpendicular given to find the hypotenuse. To find the hypotehtuse: The logarithm of ra- dius is always 10.00000. The base and perpendicular given to find the angles and the hypotenuse: To find the angles by logarithms: As base BA 15 1.17609 Is to radius 10.00000 So is perp. BC. ... 1.39794 . 11.39794 1.17609 To tang. of Ange. A 50°" '10.22185 90 Angle C. are Note.--If angle A is 59° angle C must equal what 59 lacks of 90 degrees. Angle C equals the course back, be- ing S 31°W, or SSW34W We can find angle C first if required, by changing the above proportion, making the 3d term the first term and first the 3d, thus, 1.39794 10.00000 1.17609 11.17609 1.39794 9.77915 = 31°, or 234 pts. To find the hypotenuse: As radius 10.00000 _Is to base BA 15 1.17609 So is sec. of angle A 59° 10.28816 base _ you. 11.46425 10.00000 - To the hypo. AC 29 1.46425 Note.--The above is simple propor- tion or the Rule of Three. To perform mulitplication in logarithms the logs are merely added and division is per- formed by simple subtraction. With a book of lagarithms and the foregoing explanations almost anyone ought to be equal to simple trigonometery. There will be further explanations on this subject in lessons to come, so *hat ev- erybody interested should preserve this. OBITUARIES. 'John. McNaughton, assistant lock keeper, known to every captain on the lakes, died at Sault Ste. Marie recently. Capt. Alexander A.. Johnson, eighty- five years old, one of the oldest and best known captains, died at his home - at Fennville, recently. Capt. Byron Whitaker died at his home in Detroit .recently. He was born at Hamburg near Buffalo, N. Y., and went to Detroit in 1860, enter- ing the ice business in the firm of Whitaker & Robb. He later sold out to become a vessel owner and at one time had an interest in a number of steam and sailing vessels. He had not latterly been active. He was seventy- six years old. Capt. Adelbert Krech of the Ham- burg-American liner Graf. Waldersee died at sea on May 6 of pneumonia He was born at Berlin March, 1853, and studied at the Frederick gymnasium of which his father was a director. After receiving his diploma he shipped on the Hamburg. sailing vessel and then entered the naval service at Kiel.. He made the trip to Iceland as commander of a Ger- man schooner. He entered the Ham- burg-American line service on March 14, 1883, as fourth officer and became a captain on Jan. 1, 1891. He commanded the Valdivia on her South Pole expe- dition, the Columbia, the Bluecher and the Graf Waldersee: He was to have taken command of the new Hamburg- American liner General Grant had he lived. Capt. Krech had received many decorations and presents for gallantry. Some of these are the Royal Prussian Order of the Red Eagle,- the Knight Cross of the Royal Saxonian Albrecht Order, the Knight 'Cross of the Rou- manian Crown, the Turkish Medjidi Or- der, a gold medal from the Royal Scot- tish Geographical Society, a gold ~medal from the German Emperor for rescuing shipwrecked sailors, a loving cup from President Cleveland and a diploma of honor for rescuing shipwreeked sailors.