(4) _Cases where there is no direct relation between the person, fact, or event, and the date, or number word or words._ In such cases, Synthesis, which is taught hereafter, develops an _indirect_ relation. Synthesis is used in three cases: (1) Where there is no relation _existing_ between the fact or event and its date word; (2) Where _we are ignorant_ of all the facts which would give us significant or analytic date-words; and (3) where we know the needful pertinent facts with which analytic words could be formed, but we cannot _recall_ them for use. In these three cases Synthesis must be used. I will now give and illustrate the rules for the prompt finding of _analytic date or number words_. The _preparation_ for thus remembering numbers without effort is the only exertion required. When the method is mastered, the _application_ of it is made with the greatest ease and pleasure. There are four indispensable requisites to finding analytic date and number words promptly. (1) SUCH A MASTERY OF THE FIGURE ALPHABET THAT THE CONSONANT EQUIVALENTS OF THE CIPHER AND NINE DIGITS ARE AT INSTANT COMMAND, AND NEVER HAVE TO BE LOOKED UP WHEN YOU HAVE TO DEAL WITH FIGURES. Pumps were invented in 1425. A student who thinks 2 is to be translated by "m" instead of "n," translates the dates by these phrases, _viz._, "Drum a whale," or "Trim oil," or "To ram a wall." As these phrases sustain the relation neither of In., Ex., or Con. to the fact, they are hard to be remembered; and if remembered, they mislead. The student who has mastered the Fig. Alphabet remembers that "n" stands for 2, and if he knows the object of pumps, he at once finds the analytic phrase, "Drain a well." The formula would be: "The pump invented--{D}{r}ai{n} a we{l}l (1425)," or (1) Wa{t}er (4) {r}aised (2) i{n} a (5) ho{l}low. How could he forget the date? Tea was first used in Europe in 1601. The unobserving student imagines that 6 is translated by g^hard, k, c^hard, q, or ng, and so he translates 1601 into "Ou{tc}a{st}," (1701); a mistake of 100 years, and, besides, "Outcast" is wholly unconnected with the introduction of tea into Europe. The genuine student knows that 6 is represented by sh, j, ch, or g^soft, and so he at once finds the analytic formula: "Tea first introduced into Europe--{T}ea {ch}e{s}{t} (1601)." The figure phrase bears the relation of In. and Con. to the event, and cannot be forgotten. Besides many people believe that tea helps digestion, and such persons would find an analytic date-word thus: "Tea first used in Europe--{D}i{g}e{s}{t} (1601)."