89. At those ports where, from the large number of the Candidates, it may sometimes be found impossible to look over the work on the day of examination during the office hours, an hour in the morning of the following day may be allotted for the purpose of correcting the problems, but in no case should a Candidate have his problem returned to him for correction after he has made the second attempt.

90. The paper (E‹n. 4 A.) is only for those candidates who present themselves for examination for the first time., It is intended not only to ensure on the part of the Candidate a knowledge of the meaning of a variety of terms of great use to the Navigator, respecting which much ignorance exists at present, but, at the same time, to test his handwriting and spelling.

91. In using this paper, the Examiner will place a mark against the numbers of the particular questions which he selects for answers, and not less than ten questions should be selected. The Candidate will then write against the questions so marked his definition of the terms, in a clear and legible hand, so as to prevent the possibility of any letter being mistaken. Attention should be paid to the spelling and writing of Candidates.

92. In the questions on Deviation of the Compass the Examiner should indicate by a mark against at least twelve of the questions for answers, and those indicated must be correctly answered. The selected questions should be varied frequently, and no two Candidates should have precisely the same questions. The Candidate should be furnished with sheets of blank ruled paper with instructions that he is to write only on one side of the paper, and to answer each of the questions against which a mark is placed, in a clear and legible hand, and to commence each answer by writing down the number of the question to which it relates on the side, where a margin should always be left for that purpose. In answering Question 19, besides giving a full explanation in writing the Candidates should be required to illustrate by diagrams how the poles of the magnets are placed with regard to the compass needle for correcting the semicircular deviation, as well as the heeling error, and also how the soft iron should be placed for correcting the quadrantal deviation. A Candidate for an Extra Master's Certificate will not be required to answer these questions, but wil be examined by the Syllabus.

93. In the viva voce examination, a reasonable time should be allowed for the candidate to give his answers. No assistance should be given or leading question put.

94. Particular attention should be paid to the adjustments of the Sextant, and the written answers should be given on the ruled paper in the same manner as the answers to the questions on Exn. 7. Every candidate should have a practical knowledge of what is known as the first three adjustments, and be able to read correctly off the arc, a supposed index error to be given by the Examiner as additive, as well as reading off the arc in the usual way.

95. All outstanding or minor corrections should appear in the margin of each problem paper; also on the chart papers Exa. 9 C and Exn. 9 D, and unless all these corrections appear on the papers of the Candidate they will not be considered complete.

96. In every case the Examination, whether for Only Mate, First Mate, Master, or Master Extra, is to commence with the problems for Second Mate.

97. Examiners should bear in mind that the problems to be performed are required as tests, and for the purposes of an Examination, and not for sea-going or practical purposes alone.

98. The Candidates will be allowed to work out the various problems according to the method and the tables they have been accustomed to use.

99. The rules as to the amount of time allowed are as follow:

100.

Candidates for Second Mates' Certificates of Competency must complete the whole of their Examination in Navigation in eight hours, including the time allowed for writing the definitions (Exn. 4a), the piper on the adjustments of the sextant, and the correction of all errors and oversights; but the nautical problems on Forms "Exn. 4” and “Exn. 5" must be completed within six hours, and without the Candidates leaving the premises during that period.

101.

Candidates for Oily and First Mates' Certificates must complete the whole of their Examination in Navigation in eleven and a half hours, including the time allowed for the papers on the adjustment of the sextant, and the chart, and for the correction of all errors and oversights; but the nautical problems on Forms "Exn. 4." "Exn. 5." and "Exn. 6," must be completed within six hours, and without the Candidates leaving the premises during that period.

102. Candidates for Masters' Certificates must complete the whole of their Examination in navigation in fourteen hours, including the time allowed for the papers on the sextant, the chart, and the compass deviation, and for the correction of all errors and oversights; but the problems on Forms "Exn. 4." Exn. 5," and "Exn. 6" must be completed within six hours, and without the Candi lates leaving the premises during that period.

103. Candidates after finishing the problems required of their respective grades on the first day of Examination, should proceed, until the end of that day, with such subjects as the definitions, sextant, chart, and Question 19 of Form "Exn. 7."

104. The Sumner Problem (Exn. 6a) must on no account be given out to any of the grades on the first day of the Examination, but should be the commencement of the Candidate's work on the second day, and on completion of this problem the Candidates for Masters' Certificates should proceed with the problems on Paper "Exn. 7."

105. Candidates for Home Trade Mates' Certificates must complete the whole of their arithmetic and chart rapers within five hours, and Candidates for Home Trade Masters' Certificates within six hours.

106. Candidates for Foreign-going Steamship Certificates of Competency will be allowed the same amount of time to perform their navigation, &c., as herein-before laid down for the respective grades for the Ordinary Foreign-going Certificates.

107. Punctually at the expiration of the prescribed time all the papers should be called up, whether completed or not, and if not completed, the candidate will be declared to have failed, unless the Local Marine Board or the Examiner see fit to lengthen the period in any special case. If, however, the period is lengthened in any case, the special circumstances of that case, and the reasons for lengthening the period, together with the time allowed, must be reported to the Board of Trade by the Examiners, in the column for "remarks" on the Form Exn. 14.

108. It is anticipated that but few of the Candidates for Certificates of Competency for any of the foregoing grade; will require the whole of the time herein allowed for completing their Examination in Navigation; and ample time has been prescribed, so that Candidates may perform their work in a careful, clear, and legible manner, and to the entire satisfaction of the Examiners.

109. If the Candidate passes he will receive the Form Exn. 16, which authorises the Superintendent of the Mercantile Marine Office to which it is addressed by the Examiner to issue the Certificate to the Candidate, whose testimonials, &c., will be returned at the same time. It is therefore important that the port of destination of the Certificate should be the same on both the Form Exn. 16 and the Form Exn. 2. If circumstances should make any alteration necessary, the Examiner should see that it is made in both Forms, otherwise delay in the issue of the Certificate may be caused.

SPECIAL NOTICE TO CANDIDATES.

110. The attention of Candidates is specially called to the following Regulations:III. Candidates are required to appear at the examination room punctually at the time appointed.

112. Candidates are prohibited from bringing into the examination room books, paper, or memoranda of any kind whatever. The slightest infringement of this regulation will subject the offender to all the penalties of a failure, and he will not be allowed to present himself for re-examination for a period of three months.

• A Candidate other than Second Mate who may not previously have passed an examination may be allowed the time occupied in writing his definitions on Exn. 4d, in addition to the above.

113. In the event of any Candidate being detected in defacing, blotting, writing in, or otherwise injuring any book or books belonging to the Board, the papers of such Candidate will be detained until the book or books so defaced be replaced by him. He will not, however, be at liberty to remove the damaged book, which will still remain the property of the Board.

114. In the event of any Candidate being discovered referring to any book or memoranda, copying from another, or affording any assistance or giving any information to another, or communicating in any way with another, during the time of examination, he will subject himself to all the penalties of a failure, and he will not be allowed to be examined for a period of six months.

115. No Candidate will be allowed to work out his problems on a slate or on waste paper.

116. No Candidate will be permitted to leave the room until he has given up the paper on which he is engaged.

117. Candidates will find it more convenient, both here and at sea, to correct the declination and other elements from the Nautical Almanac by the "hourly differences" which have been given in that work in order to facilitate such calculations; they will thereby render themselves independent of any proportional or logarithmic table for such purpose.

118. The corrections by inspection from tables given in some of the works on navigation will not be allowed (see Tables IX, XI, and XXI, in Norie's Epitome, &c.); every correction must appear on the papers of the Candidates.

119. Candidates are expected to work out their answers to all problems, except those referred to hereafter, to within, or not to exceed, a margin of one mile of position from a correct result, with the exception of the ship's position by "Sumner's method," where a margin of 2 miles may be allowed.

120. In finding the longitude by chronometer the logarithms used in finding the hourangle should be taken out for seconds of arc. Also accuracy should be observed in computing the Hour-angles for the projection by "Sumner's method."

121. In all other problems the logarithms to the nearest minute will be sufficiently correct for all grades except Extra Master, from whom a degree of precision will be required, both in the work and in the results, beyond that demanded from the inferior grades.

122. Any Candidate who may be guilty of insolence to the Examiner or of other misconduct will render himself liable to the postponement of his examination, or, if he has passed, to the detention of his Certificate for such period as the Board of Trade may direct. 123. For rules as to amount of time allowed to perform the work, see "General Instructions to Examiners and Candidates" on page 12.

MASTERS' AND MATES' VOLUNTARY EXAMINATIONS IN STEAM. 124. These examinations are limited to Masters and First or Only Mates who are possessed of or entitled to Certificates of Competency, and were provided for the purpose of giving them an opportunity of undergoing a voluntary examination as to their practical knowledge of the use and working of the steam engine. They are conducted on the premises, and under the superintendence of the Local Marine Boards, at such times as they may appoint for the purpose; and the Examiners are selected by the Board of Trade from the Engineer Surveyors appointed under the fourth part of "The Merchant Shipping Act, 1854."

125. Any Master or Mate desiring to be examined in Steam must deliver to the Superintendent of the Mercantile Marine Office a statement in writing to that effect, upon the Form of Application (Exn. 2) and the applicant's Certificate of Competency must be delivered to the Superintendent along with his statement. If he is about to pass an examination for a Certificate of Competency at the same time, the applications should be sent in together. 126. A fee of one pound must be paid by the applicant for the examination in Steam, and the Superintendent of the Mercantile Marine Office will thereupon inform him of the time and place at which he is to attend to be examined, and the examination will then and

there proceed in the same manner as the other examination. If the applicant fails, and has given in his certificate, it will be at once returned to him.

127. If the applicant passes, the Report (Exn. 14) will be sent to the Board of Trade, and the Certificate of Competency with the Form (Exn. 2) to the Registrar-General of Seamen; the words "Passed in Steam" with the date and place of examination, will then be entered on the Certificate and its counterpart, and the Certificate will be sent to the Superintendent of the Mercantile Marine Office of the port named in the Application (Exn. 2) to be delivered to the applicant in the usual manner.

128. If the applicant fails he may not present himself for re-examination until the expiration of three months from the date of failure.

129. The examination is for the most part viva voce, and extends to a general knowledge of the practical use and working of the steam engine, and of the various valves, fittings, and pieces of machinery connected with it. Intricate theoretical questions on calculations of horse-power or areas of cylinders and valves, or any of the more difficult questions which appertain to steam engines and boilers, will not be asked. The examination will in fact be confined to what a Master of a Steam vessel may be called upon to perform in the case of the death, incapacity, or delinquency of the engineer.

130. Examiners are to be careful in their examinations to satisfy themselves that applicants really do know the names and uses of the various parts of engines and boilers, and their connecting pipes, valves, cocks, &c. Practical knowledge, as distinguished from theories, abstruse calculations, and book learning, is to be the test of the applicant's fitness to have his certificate indorsed.

131. The Examiner should arrange to conduct part of the examination in the engine room of a Steamship, unless from circumstances he finds it impossible to do so; and if an opportunity offer the applicant should be permitted, under the guidance of the Examiner, to start and stop the engine of some vessel which may have her steam up.

132. The Examiner in sending in his report of Examinations of Masters and Mates in steam, should state where the examination has been held.

133. Candidates will be required to give written answers to sixteen out of twenty questions taken from a book of Elementary Questions published for the Board of Trade.* These questions will be altered from time to time without notice. The twenty questions are not to be difficult, theoretical, or book questions, but are to be such as any man of ordinary capacity ought to answer who has any "practical knowledge of the use and working of the steam-engine."

134. These questions with the Candidate's answers, should be sent to the Board of Trade, with the reports, after each examination.

135. These answers will also assist in enabling the Board to decide any question that may hereafter arise in cases where it is alleged that applicants have been improperly passed, or improperly rejected.

136. The Examiners will be careful that if any candidate refers to any book or paper or memorandum, or obtains information from another candidate, during the examination, he will be treated as having failed, will forfeit his fee, and will not be allowed to be re-examined for a period of three months.

137. The Examiners will report in case of failure, the nature of the question or questions that decided the failure, or the point in the management of the engine in which the candidate was deficient.

138. There is nothing in the regulations requiring that applicants for the voluntary examination shall have served on board Steam Ships; all that is required is that they shall nave a "practical knowledge." Examiners will, of course, not fail to appreciate the fact, when passing applicants, that practical knowledge is best gained in the engine room; and the examination of an Officer who does not produce official evidence of service in Steam Ships, and of experience of engines, must necessarily be more searching than in the case of one who produces evidence of such service and experience.

For the Answers see "AINSLEY'S ENGINEER's Manual.”

ON LOGARITHMS.

1. Logarithms are numbers arranged in Tables for the purpose of facilitating arithmetical computations. They are adapted to the natural numbers, 1, 2, 3, .... in such a manner that by means of them

the operation of Multiplication is changed into that of Addition;

2. Take any whole numbers, as 18, 813, 6489; the first consists of two, the second of three, and the third of four figures or digits. Again, in the mixed number 739.815, the whole number or integral part (739) consists of three digits.

3. By multiplying a number by itself, one, two, three, &c., times successively, we obtain the second, third, fourth, &c., powers of that number; hence, a power of a number is the number arising from successive multiplication by itself. Thus, 3 x 39 is the square or second power of 3; and 5 x 5 x 5 = 125, the cube or third power of 5; and so on.

These operations are denoted by means of Indices, or small figures placed on the right of the numbers, a little above the line; thus, 2 = 2 x 2 = 4, 3o = 3 × 3 × 3 = 27, and 25 = 2 X 2 X 2 X 2 X 2 = 32, where the Index or exponent denotes the number of factors employed.

4. When there are series of numbers, such that each is found from the previous one by the addition or subtraction of the same number, they are said to be in arithmetical progression. 1, 3, 5, 7, 9, 11, &c., are in arithmetical or equi-different progression, since each number is found by adding 2 to the immediately preceding.

No proof can here be offered that numbers must exist possessing the properties under which we call them logarithms; neither can any account be here given of the methods of computing such logarithms. The reader will accept the statements that if such numbers exist, bearing the properties aforesaid, they are called logarithms. He must also accept the tables which are published, recording logarithms for the several numbers to which they profess to belong, though he cannot at present verify the computations of these several logarithms; and he will be informed how he may use these tables to effect with comparative ease many calculations which would otherwise be most laborious.

The truth is, though it requires for its demonstration higher algebra than this work presupposes the reader to be acquainted with, that not only has every number a logarithm, but it has an infinite variety of logarithms, constructed, as the term is, on different scales or bases. The base of any system of logarithms is defined by the fact that in that system unity is its logarithm.

Any number might be used as a base; but in fact there are only two numbers which are ever really used.

The one is an unterminating decimal, 2.7182818.. denoted generally by the letter e. This is the base of what is called the natural or Naperian system; and the advantage of it consists in the ease with which logs. are computed to this base, but which we cannot here explain.

The other is 10, which is the base in ordinary use, and with this base log. 101. Logarithms to this base are the only ones which will now be considered in their practical use.

D