PRINTING

since 1910 many improvements have been made in high speed machinery and in mechanical details. linotvpe.—the linotype machine has been improved and is made with four main magazines, each containing a complete range of types of a different body or face. a lever changes the magazine, and the width of the mould can be altered quickly so that varying types and sizes can be set without delay. side magazines, worked by an additional keyboard, are fitted, which will set much larger type on the normal body, the letters over- hanging aad being supported by leads or slugs. ifouotype.—the monotype has a greater range of faces which can be set at one time. the machine will set solid type up to 24 points at a much slower speed, and also overhanging initial letters which are larger than the normal body. the improve- ments on both machines are very useful for catalogue and news- paper work. tire ludlow.—in the ludlow the matrices of the type are set by hand, and clamped in the machine, which casts a solid line of type, and the matrices are returned to the case by hand. this machine is used entirely for display work or advertisements, and large types, with an overhang supported by slugs, can be cast. a wide range of faces and sizes in unlimited quantities, or repetition lines, can thus be obtained, and the type is always cast afresh for each job. the supertype composing machine matrices are set in a manner similar to the linotype, but single letters are cast instead of a slug for each line. automatic feeder —the use of automatic machines to feed the sheets of paper to the printing machine has been widely adopted, and continuous running is possible for long periods, whereas with hand-fed sheets it is necessary to stop the machine frequently when a fresh supply of paper is put on the feed board. mechanical feeders are of the “pile” or the “ continuous ” type. in the pile feeder a stack of paper is automatically raised as each sheet is separated and lifted by air suction, and passed down a sloping board to front lay marks, and adjusted by mov- ing side marks which place the sheet in the exact position for the grippers on the printing cylinder. electrical or mechanical devices stop the machine if two sheets or no sheet be fed in by error. in the continuous type of feeder, small quantities of paper are placed and combed out by hand on the feed board, and then mechanically fed to the machine by somewhat similar means. with these mechanical feeders a higher output is ob- tained, and in large sizes a stack of paper can be wheeled on a trolley to the feeder. an extended delivery carries the printed sheets beyond the machine to a delivery board which descends gradually, and a truck load of paper can be whecled away with- out handling to the binding department, and thus much handling and time is saved. the smaller sizes of printing machines are now built with an automatic feeder as an integral part of the construction, and speeds up te 4,000 per hour are possible, which could not be attained on a hand-fed machine. process work.—three-colour printing is used in increasing quantities for hlustrations ef all kinds. absolute accuracy in register, height to paper, etc., is essential, and many improve- ments have been made. blocks before being printed are gauged to 1,000th of an inch, and underlaid to make them dead-level. if a large cdition be required, and plates are used on a flat bed printing machine, stecl mounting blocks, planed to 1,oooth of an inch, built up in small sections, with screws for adjusting the position of the plates, give a ilat hard printing surface and per- fect register between the three or four colour plates. less time is then taken in making ready, and with modern machinery long runs of colour work can be obtained at a high speed without showing bad register or signs of wear. good colour printing requires uniform liphting, and special electric larnps have been perfected to give the effect of sunlight er northlight, and so enable printing to be carried on day or right without the variation caused by changing from natural to ordinary artificial hght. the expansion and contraction of paper caused by variations in temperature or humidity prevent good register, essential in colour printing. self recording hygrom- eters and thermometers are used to overcome these difliculties. z20 newspaper work.—in newspaper printing there has been a steady improvement in details. most of the daily papers now have a page devoted to half-tone illustrations, which were for- merly restricted to the daily papers specialising in pictures (sec newspapers). the growth in the circulation of the newspapers has been aided by the increased output of the modern newspaper printing machine. the running speed of the machines is much higher, and the following are the principal improvements. roller bearings have been fitted to the printing impression cylinders, and ball bearings to some of the other moving parts, in order to reduce friction and give smoother and easier running. for the inking rollers, which are covered with a rubber-like coating made of glue, treacle and glycerine, a new non-melting compo- sition has been invented which enables the machines to run at a higher speed. a fine spray of ink is forced under pressure through a number of minute holes upon the rollers, supplying a continuous and uniform quantity of ink when the machine is running at a high speed. costing methods —mention may be made of the development of proper costing methods in the printing trade. this movement began in the united states and spread later to great britain, with the result that in both countries a uniform system has been officially adopted. the system leads to many improvements, and points out the high cost of running obsolete and inefficient machinery, and where losses are incurred. the methods of dealing in paper have been standardised, first in great britain and afterwards in the united states. the sizes have been fixed and the ream containing a variable number of sheets abandoned in favour of 1,000 sheets as a basis for selling, which gives the advantage of decimal calculations. other investigations are being made in order to standardise and simplify the methods which have been the slow growth of centuries. (we es) presses of various kinds styles of presses -—your styles of presses were in common use in tg910 in book and job offices: platens, drum cylinders, siop cylinders and two-revolution cylinders. in the platen press the form was held in a_ perpendicularly stationary position, the printing surface, or “ platen,” opening on a hinge movement during the feeding of the sheets and while the inking rollers were passing up and down over the surface of the type forme. the drum-cylinder press was so called because of the resemblance of the end view of its impression cylinder to a large drum. enough of the circumference of the cylinder was cut away to permit, after the impression had occurred and without stopping the cylinder, of the return of the type forme for re-inking and another impression. in the stop-cylinder press, only a small part of the circumference of the cylinder was cut away; the cylinder, which was of course smaller in size than in the drum, stopped with its flat side at rest over the forme, which thereby was provided with sufficient headroom for its return. in the two-revolution press the cylinder was not stopped but was automatically raised and, while the forme was returning, the cylinder made a second revo- lution. the motion lost by the stopping of the cylinder in the drum- and stop-cylinder presses was saved in the two-revolution press, thereby giving it greater speed and consequently more production. printing of newspapers of all but limited circulation was done on rotary presses. the forme was stereotyped in a half-circle, two pages encompassing each cylinder, which in printing rotated against the impression cylinder, lost motion being avoided and the ultimate in speed seemingly achieved. vhe presses were built on the “ deck ” system, one unit above another. the paper was fed from a roll at one end and came out at the other as com- pleted and folded newspapers. in lithographic printing the stone carrying the printing surface was just beginning to give way to zinc and aluminium sheets for the same purpose. “ straight matter”? composition was done on two type-casting machines, one producing single types and the other producing completed lines or ‘ slugs.” the situation in 1926 was as follows:— printing the platen press was still much used. for short runs it con- tinued to be hand-fed; for long runs it was equipped with auto- matic feeding devices that greatly augmented production such as those described above. both drum-cylinder and stop-cylinder presses had practically disappeared. the two-revolution press had been developed to a high state of perfection; it was built in small and large sizes and also for printing in two colours at one operation. | automatic job presses have come into general operation since 1910. three of them are the kelly, which embodies the principle of the two-revolution cylinder press; the miehle vertical, which is in effect a fast small stop-cylinder with a vertical stationary form; and the miller high-speed press, which has a two-revo- lution movement with the diiference that the bed and not the impression cylinder moves up and down. rotary presses.—the rotary principle still prevails in news- paper offices, but the “deck ” style of construction has been abandoned for what is called the ‘‘ multiple unit ”’ basis. under this system, when additional equipment is required one press is not superseded by another; a unit is simply added and thereby ° a sextuple becomes an octuple, an octuple a decuple, etc. the presses are built heavier and stronger. the old method of hand stereotyping has been superseded by autoplating. a further development in press construction, known as the “ tubular ” system, has been perfected. by its use each plate is cast as a tube instead of in semi-circular form. this tube is slipped over a cylinder, which prints a complete page with every revolution. the rotogravure press (see puoro-engraving) is making rapid strides in the newspaper field, and) may be expected in time to play an important part in the commercial and book printing ficlds also. most illustrated sunday supplements are printed on rotogravure presscs. lithography.—in lithographic printing, the stone from which impressions were formerly taken has almost completely disap- peared. in its stead are zinc and aluminium sheets which, en- circling the printing cylinder, permit the complete application of the rotary principle to lithographic production also. a further development is what is known as the “ offset ’ process, in the operation of which the design to be printed is first transferred to a rubber blanket, from which it is again transferred to the paper. whereas formerly only smooth-surfaced, specially prepared papers could be printed upon lithographically, by the offset method any paper, whether smooth- or rough-surfaced, may be used with equal facility. offset seems to be particularly adapted to colour work. tonnage production—a new field, sometimes known as “tonnage production,” concerns itself with magazine printing and is the outstanding development in printing machinery during the 15 years ending in 1926. whereas only a few years earlier a circulation of 500,000 copies of weekly and monthly periodicals, printed in one colour, was a rarity, in 1926 four and five times that number, with almost numberless pages in two and four colours, is accepted as nothing out of the ordinary. offsets -—tonnage production in one colour presented only a single problem, which was that of quickly drying the sheets. the prepared surface on a cylinder against which a sheet is pressed in the impression contact is called a tympan. when a sheet is printed on both sides during a continuing operation, the printed side is likely to smudge the tympan as the blank side is taking the impression: this smudge when it appears on the next printing is known as “ offset.” when properly controlled, as in lithographic printing, offset becomes a process and produces fine effects. in relief printing, on highly surfaced papers, how- ever, ofisct must of necessity be eliminated. offset 1s not a prob- lem in newspaper production, for the reason that printing on newsprint dries quickly, but the case is different with printing on supercalendered and coated papers, such as must be employed for the half-tone and process plates with which magazines are illustrated. at first the remedy was believed to be found in a “ shifting tympan ” that changed the surface at intervals, but the device was abandoned because of the time required for the shifts. . printing machinery plate fic. 1. goss unit type octuple press 174-a, for printing newspapers and delivering them (centre of picture) folded to half-page size and counted; size of page usually about 17x23 inches. l speed of 72,000 copies, 16 pages, per hour, or 36,000 copies, 32 pages. requires 75 horsepower. fic. 2. kelly press, model 2; prints paper from 8x12” to 22x34” at 2,000 to 3,000 impressions per hour. the automatic feeder (at leftend of picture) holds a stack of paper up to 213”. the press delivers the printed sheets into a jogger-box or layboy, the bottom of which is a truck (right end of picture). as the load increases the truck is lowered, and the fully loaded truck is then hauled away without further handling of the paper. fg. 3. miechle vertical press; prints paper from 3$x5}” to 12}x19” at from 2,500 to 3,600 impressions per hour. requires 2 horsepower. fic. 4. cottrell multi-colour sheet-feed rotary press, printing 4 colours simulta- neously on one side of the paper, in sheets up to 47x69” at 2,700 to 3,000 copies per hour. it prints from curved electrotypes. prisoners of war the cottrell rotary press—the late mr. c. b. cottrell, founder of c. b. cottrell and sons co., solved the problem with the cottrell rotary press. it provided an automatic shifting tympan that furnished an entirely new and clean impression surface as often as necessary to prevent offset. this press, with roll feed to print on both sides of the web, had all the fine print- ing qualities and conveniences of the flat bed press, plus the speed of the rotary, and met at high speed all the requirements of printing without smut or offset. the shifting tympan has been superseded by a “ travelling offset” web. the cottrell co. added colour cylinders to the printed units and so perfected two colours on each side of the web, and, in subsequent develop- ments along the line of multi-coloured printing, has produced a rotary perfecting press, printing from a roll of paper and produc- ing four colours on one side and two or four colours on the other side of the web. a further improvement is the mckee process for colour snake ing, which produces a printing plate having graduated eleva- tions on its printing surface, corresponding in order with the solid or dark tones and the intermediate tones of the subject on the face of the plate, varying in structural thickness in accord- ance with the several pressures required for printing and in accordance with the vazious tones of the subject on the face of the plate. such a printing plate has the sections which are designed to print the darker shades permanently elevated above the levels of the sections adapted for the lighter shades, and the levels are graded one into the other from the permanently cle- vated parts to the lightest printed shades, so that in profile the plate is uneven or irregular in its printing surface and level on the back. the usual underlays and overlays are not required, since their effects are produced on the face of the plates, and the ‘* strike-off ’? between the cut overlay and plate which happens so often is obviated. in short, a printing plate is produced wherein the required make-ready is permanently produced in the face of the plate. until the mckee process was perfected and used in conjunction with the cottrell multi-colour press, it was customary to print only one colour at a time at each passage of the sheet through the press, requiring as many separate and independent printings as there were colours required. by the mckee process, colour printing can be produced on presses having one impression cylinder common to all of the plate cylinders. the cottrell multi-colour press produces by a continuous operation that consists of one passage of a sheet through the press a picture in which the different colour values are effectively brought out. see g. t. jacobi, printing (1919); j. southward, afodern printing (1922 (jg. 0) prisoners of war.—the procedure laid down by inter- national agreement for the treatment of prisoners of war under the hague regulations was severely tested during the world war. in the first place, it must be borne in mind that great britain, france, italy, germany and austria together claim to have taken not less than 6,000,000 prisoners, while austria admits having lost 1,500,000 prisoners to russia. of the 200,000 british prisoners of war more than 90%) were taken by the germans, the remainder being principally in bul- garian and turkish hands. the following brief account will, therefore, deal for the most part with the treatment and experi- ence of british prisoners of war in germany and conversely of german prisoners of war in great britain. tor this purpose three divisions will be made, namely, civilian or non-combatant prisoners; officers; and rank and file. civilians.—the internment of large numbers of civilians had not been contemplated by the framers of the hague regulations, and therefore presented an entirely new problem. by the end of nov. 1914, some 29,000 germans were interned in great britain, while in germany more than 112,000 civilians, of whom not more than 6,o00 were british, were in a similar position. in both countries accommodation was at first definitely bad. newbury race-course was one of the first internment camps in england. the german reply to this was ruhleben race-course. newbury as | was abandoned after a few weeks, but ruhleben remained the principal civilian camp till the end of the war. decidedly bad at first, conditions there gradually improved, largely as a result of the efforts of the prisoners themselves. civilian prisoners were in neither country cornpelled to do any work beyond what was necessary for the cleanliness and order of their camps. in great britain, however, civilian prisoners were given the opportunity to volunteer to work, and some 1,500 were given employment, mainly in agriculture. in great britain each internment camp had its own small hospital. at ruhleben the medical arrangements were at first quite inade- quate, but later they improved considerably. officers. —the treatment of all combatant prisoners in ger- many, officers and men alike, varied greatly with the situation of the camp, and particularly with the personality of the com- mandant. in the x. army corps district, for example, all pris- oners came under the malign influence of gen. von hiinisch, who invariably appointed men of his own kidney as camp comman- dants. accommodation ranged from comfortable hotels, as at augustabad, to badly constructed huts, as at strohen, near han- over. fuel was always scarce, in some cases, as at holzminden, practically non-existent. as regards food, the insufficiency and unsuitability of the german rations made the prisoners almost entirely dependent on parcels from home. it is to the credit of the german authorities that the majority of these parcels arrived intact. no work was required of officers; and for the care of the camp, orderlies, usually of the same nationality as the officers, were provided in most cases. the most serious complaints that can fairly be made on hehalf of officer prisoners in germany are shortage of fuel, inadequacy of cooking arrangements, and especially in the x. army corps district, the determination of the camp comman- dants to prevent the prisoners from introducing any degree of comfort into their situation. german officers were confined in england in large country houses and public institutions. it can fairly be said that on the whole they were better fed and housed than british officers in germany, and the facilities provided for their recreation were much more adequate. rank and file-—while the experiences of civilian and officer prisoners of war are of considerable interest, it is undoubtedly by their treatment of the rank and file prisoners that the nations concerned must be judged. all belligerents compelled prisoners to work, as they were entitled to do, provided that prisoners were not compelled to undertake work directly connected with the military activities of their captors. according to numerous re- ports, all the belligerents at various times broke this rule in the spirit at least; and germany several times broke it in the letter. camps were of tivo principal classes, main camps and working camps. the former were usually composed of hutments similar to those scen in the english temporary military camps, the whole being surrounded by barbed wire, and frequently sub- divided by barbed wire into sections. the outstanding complaint from british prisoners regarding the main camps in germany seem to be on the ground of overcrow ding—at wittenberg, for instance, 15,000 to 17,000 men were confined in an area of about 1o acres. as regards accommodation working camps varied enormously. men sent to work in factories were sometimes housed m1 quarters normally provided by the employer for bachelor workmen; men working on farms lived in small camps erected near the farm or occasionally in the farm buildings them- selves. treatment in such quarters likewise varied with the personality of the employer or of the non-commissioned officer in charge of the camp or compound. on the whole the german non-commissioned officer was a harsh and frequently brutal commandant. the discipline in german factories, mines and the like was at all times very severe; and apart from disciplinary measures, too many authentic cases of gratuitous brutality have been recorded against the german authorities, particularly against non-commissioned officers in charge of working parties. reprisals.—hard though the lot of the prisoners of war set to work in factories and mines undoubtedly was, the most 222 terrible chapter of the sad story concerns those unfortunate individuals selected as the subjects of reprisals by their captors. this barbaric method of avenging wrongs inflicted on their sub- jects was employed by all the belligerents from time to time, but authoritative evidence shows that germany was in this respect by far the worst offender. one of the most notorious examples of german reprisals was the transfer of 500 british soldiers to the eastern front in feb. 1917. after being forced to march long distances through appalling country, with insufficient food and clothing, these unfortunates were housed in a tent pitched on the snow beside the frozen river aa, within range of the russian guns. many of the men died and many others lost fingers and toes through frost-bite. organisations for prisoners.—the relief of prisoners of war was at first left in the various countries to individuals. it consisted inevitably in the sending of parcels of food and other necessaries. in great britain, whose subjects were much worse off than were the german prisoners, it gradually became clear that some con- trol of the parcel supply was required, as individual enterprise resulted not only in an unfair distribution of parcels, but also jn an imperfect selection of commodities to be sent. early in rors, a prisoners of war ilelp committee was established in london. having no authority, however, it was a comparative failure, and in sept. 1916, it was superseded by the central prisoners of war committee of the british red cross and order of st. john. this body had absolute control of parcels and from its inception till the end of the war maintained a steady stream of suitable parcels to all ranks. repatriation.—by aug. 1915, the british and german govern- ments had reached an agreement regarding the repatriation of officers and non-commissioned officers incapacitated by wounds or illness from further active participation in the war. further agreements were subsequently reached providing that prisoners less seriously wounded or ill, and prisoners who had been in captivity for over 18 months, might be sent to holland or switzerland for internment there. immediately after the armi- stice, the repatriation of all prisoners commenced, but owing largely to transport difficulties it was several months before the undertaking was completed. turkish and bulgarian prisoners british soldiers who were captured by the turks have brought very serious charges against their captors. officers, particularly those of high rank, appear to have been treated with almost theatrical courtesy; but the rank and file were subjected to appalling hardships, being housed in quarters which were nothing short of filthy, and fed and clothed in a manner deserving of the severest censure. many of the prisoners died and many more returned after the war com- pletely broken in health. the bulgarians, once they understood that the somewhat primitive disciplinary methods employed in their own army were quite unsuitable and unnecessary in the case of british soldiers, treated their prisoners with quite a reasonable degree of human- ity, and fed, housed and clothed them as well as could be ex- pected, having regard to the resources at their disposal. bibliography.—a large number of reports were rendered at various times by the united states on the treatment of prisoners of war in great britain and germany; and a considerable amount of correspondence took place between great britain, germany and turkey. of non-official works, j. w. gerard's afy four years in germany (1917) is perhaps the most reliable. for a picture of life as an officer prisoner in turkey see fe. ii. jones, the road to en-dor (1920). (c.d) probability (sce 22.376).—1. the relation of the mathemat- ical theory of probability to other branches of science has al- tered to some extent. from being a separate subject, providing interesting problems to the mathematician, it has almost come to bea part of a larger subject, the theory of statistical frequency; at any rate it is one of the aspects in which statistical frequency may be studied. the theory of probability was originally con- cerned mainly with games of chance, and to some extent also with vital statistics. the discovery of the “law of error” brought all scientific observations into its field. and it has probability and correlation, mainly arising out of francis galton’s investiga- tions of heredity. this has led to some reconsidcration of principles. 2. typical questions.—a box contains 5 balls, all, so faras we can see, exactly alike, except that one is red and the remaining 4 are white. a ball is drawn at random from the box. what is the probability that it will be the red ball? a reasonable answer, but a provisional one (see para. 8), is that the probability is 1¢. but, before we give this answer, the question itself gives rise to several questions. what do we mean by “at random?” what do we mean by “ probability?” in what sense can a probability be a number, or be measured by a number? why does the number necessarily lie between o and tr? how is this number, which is or measures the probability, to be ascertained? and at what stage of the proceedings does probabil- itv, as such, cease to exist? we say that a ball “ is drawn;” do we mean that it is going to be drawn, and that the probability that it will be the red ball ceases to exist as soon as the ball is drawn? or, supposing the ball has been drawn and we do not know which ball it is, can we say that the probability that it is the red ball is so-and-so? 3. afeusurement of probability—for numerical measurement of probability there seem to have been three methods used; cach of them leads to 1 as the measure of certainty. (i) the study of probability was originally, in the main, the scientific study of gambling. if <1 has a chance of winning a prize x, the primary object was to find ¥, the value of his chance. the ratio of vito x isa fraction between oand 1, which may be called the probahil- ity of a winning the prize. in the present case, suppose there are § persons, each of whom in turn is to draw a ball, and that whoever draws the red ball is to get 10 shillings. then it will no doubt be agreed that each person’s chance of drawing the red ball is worth 2 shillings; each person’s probability of drawing this ball can there- fore be said to be 44. gi) the rule usually given in text-books is as follows: if an event can happen in n ways, all of which are equally likely, and if p of these ways are called ‘ favourable,’’ then the probability of the event happening favourably is ?/n. one objection to this ts that it introduces a new phrase, “ equally likely,’’ without defining it. if what is meant is that the numerical probabilities are equal, we are working in a circle. another objection is that it does not cover all cases. : (ii) the modern idea is to base the probability on statistics. to find the probability that under conditions can event fe will happen, we enquire as to the relative number of cases in which it actually dves happen. if we find, or if it is reasonable to suppose, that in .v cases in which the conditions c are created it happens in pv cases, we can say that the probability of it happening under these condi- tionsis p. it is assumed, of course, that its happening or not happen- ing in any one case is not affected by its happening or not happening in any other case. the adoption of the statistical basis disposes of a good many difficulties. jf, for instance, the ball has been drawn, so that it is definitely red or white, but we do not know which, we can speak of the probability that it is red; for we are dealing statistically with the class of cases in which the ball has been drawn, and are considering, out of these, the cases in which the ball is red. 4. randomness.—the expression “‘ at random,” or random- ness, does not seem, at any rate on the statistical basis, to mean anvthing more than that we are dealing with a question of chance. jo say “ a ball is taken at random, the chance that it will be red being 1%” is the same as to say “a ball is taken under such conditions that the chance that 1t will be red is 1.” s. mathematical treatment.—the treatment of a proklem in probability usually consists of three stages:—(1) the numerical determination of probabilities, either of single events or of joint occurrence of events (2) the deduction, by mathematical methods, of other probabilities (3) the application of the results so ob- tained. we have so far dealt with (1). we have now to consider (2). the mathematical treatment involves the addition and multiplication of probabilities, and the inverse processes, sub- traction and division. ‘these processes follow certain rules. g) the addition rule is that if an event c (e.g., the drawing of a ball from a box) can be associated with one or other of two events flond f (eve, the ball drawn being red or being white) but not with f * | both, and if the probability of cbeing associated with / is p, and the i srcbability of c being associated with f is q, then the probability of received a great stimulus frorn the statistical study of variation | c being associated either with e or with f is p+. probation .- (ii) the multiplication rule is that if the probability of c being associated with e is p, and if, in the cases in which c is associated with e, the probability of d being associated with g is r, then the probability that c will be associated with 4 and d with g, is pr. (this form of statement avoids the introduction of *‘ independence ”’ of probabilities.) these rules are sometimes stated as theorems, sometimes as axioms. they are both deducible from the statistical definition given in (iii) of para. 3. 6. inverse probabilitv.—a class of problems of increasing im- portance is that of determining the probability that an event & has been preceded by an event c, this being one of a definite number of possible and mutually exclusive antecedent events c, cc’, c” ... . such a probability is often called an zaverse or a posteriori probability; the probability of an event c being followed by an event £ being called a direct or a prior? probabil- ity. the treatment of “ inverse probability ’’cases has sometimes been discredited, not without justification; the problems which mathematicians have sct have often been impossible of solution, on account of the data being insufficient, and arbitrary assump- tions have had to be made. the following may be taken as a typical question in inverse probability. a box contains 5 balls, 1 of which is known to be white and 3 red, the remaining one being either white or red. a bail is drawn and is found to be white. what is the probability that the box contained 2 white and 3 red balls? it is impossible to answer this question without further information. the box has been filled in some way: the method of filling having led to the box containing i white and 3 red balls, we require to know the probability that it will lead to the other ball being white or being red, respectively. when we know this, the problem is a straightforward one, if we use the statistical method. suppose that the probabilities in question are 3g and 3¢ (as might possibly be the case if the box were filled from a source containing white and red balls in the ratio of 3:4). then we may express the problem as follows: there are ‘“w" boxes containing 2 white and 3 red balls, and ‘‘ r ” boxes containing 1 white and 4 red balls; their frequencies being in the ratio of 3:2. a box is taken at random; and a ball is drawn from it and is found to be white. what is the probability that the box isa w box? from the data we get the fol- lowing (table i.) as a representative frequency-distribution for 1,000 such drawings: — table lf. from w boxes from all boxes from r boxes white ball drawn red ball drawn . 240 80 360 320 320 680 600 i ,ooo total drawings 400 it will be seen that the required probability is 240/320= 7. (this can be verifed by using the ordinary method.) now compare the above question with the following. a man being called short or tall according as his height is under or over 674 in., suppose that, in a community in which every man has one son who attains maturity, the statistical relation between height of father and height of son, in a representative i,000 pairs of father and son, is given in table ii.:—! table it. father father total short tall fathers son short 250 89 339 son tall 215 alg 661 total sons 465 535 1,000 what is the probability (@4) that the son of a tall father is tall, (0) that the father of a tall son is tall? the answers are (¢) 446/535 (b) 446/661. if we make a distinction between ‘ direct ” and “inverse ’ probability then, since the father precedes the son, («) is a case of direct and (6) a case of inverse probability. but in which category are we to put such a question as:—what is the probability that a girl with fair hair has blue eyes? it is clear not only that there is no fundamental difference between the two classes of cascs, but that, if we do make a distinction, we must introduce a third category, that of probabilities with regard to concurrent events. 9. errors of randomness.—the distribution in table i. is a representative distribution, t.e., the numbers in the different com- partments of the table are exactly proportional to the corre- sponding probabilities. but, if 1,000 boxes were taken at random in the way described and balls were drawn at random from 1 adapted from k. pearsonand a. lee, biometrika, vol. ii (1903), 415. 223 them, these exact numbers would not be likely to appear. the differences between the actual numbers obtained and these theo- retical numbers are the errors of randomness, or errors of random sampling. the consideration of the relative frequencies of errors of this kind constitutes the theory of errar, which is the main link between the theory of probability and the theory of statistical frequency. it forms a large part of the former, and is a par- ticular case of the latter. 8. lack of finality—we may revert, finally, to the question put at the beginning of para. 2. we said that, on the data, we may reasonably say that the probability of the red ball being drawn is 4%th. but this is only a provisional estimate, the test of which is the result of actual drawings. if we find that the red ball is drawn more often than each one of the white balls, we must conclude that our original estimate of the probability was wrong, and must modify it accordingly. this is, in effect, using the statistical basis. but even this basis only enables us to obtain an approximate value, which is liable to be modified by further statistics. it is never possible to find the true value of any particular probability. and the problem of determining the best value, being based on an investigation of past facts, is really a problem of inverse probability. references.—for a quite elementary view of the theory of probability, reference may be made to w. a. whitworth’s choice and chance, or to chapters in some work on algebra, such as those of g. chrystal or h. s. hall and s. r. knight. the principles of the subject were considered by j. venn, the legic ef chance (ist ed. 1866); this ts still the standard work. j. m. keynes, al treatise on probability (1921), deals with the logical foundations of probability, whether capable of numcrical measurement or not. on the statistical side, g. u. yule, intro. to the theory of statistics (1911), is of value, especially as regards the theory of sampling. a. l. bowley, elements of statistics (4th ed. 1920), is more advanced. a very full mathe- matical equipment is provided by e. t. whittaker and g. robinson, the calculus of observations (1924). a considerable number of books on the mathematical study of statistics have appeared. the annua! volumes of biometrika, from 1902 onwards, are indispensable to the serious student. 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