The biggest and cleverest mechanical brain of the analogue type which has yet been built is the _differential analyzer_ finished in 1942 at Massachusetts Institute of Technology in Cambridge, Mass. The fundamental physical quantity used in this machine is the amount of turning of a shaft. The name _analyzer_ means an apparatus or machine for analyzing or solving problems. It happens that the word “analyzer” has been used rather more often in connection with analogue machines, and so in many cases the word “analyzer” carries the meaning “analogue” as well. The word “differential” in the phrase “differential analyzer” refers to the main purpose of the machine: it is specially adapted for solving problems involving _differential equations_. Now what is a differential equation? DIFFERENTIAL EQUATIONS In order to explain what a differential equation is, we need to use certain ideas. These ideas are: _equation_; _formula_; _function_; _rate of change_; _interval_; _derivative_; and _integral_. In the next few paragraphs, we shall introduce these ideas briefly, with some explanation and examples. It is entirely possible for anyone to understand these ideas rather easily, by collecting true statements about them; no one should feel that because these ideas may be new they cannot be understood readily. PHYSICAL PROBLEMS In physics, chemistry, mechanics, and other sciences there are many problems in which the behavior of distance, of time, of speed, heat, volume, electrical current, weight, acceleration, pressure, and many other _physical quantities_ are related to each other. Examples of such problems are: [Illustration: FIG. 6. Paths of a shot from a gun, trajectories.] What are the various angles to which a gun should be raised in order that it may shoot various distances? (See Fig. 6.) (The paths of a shot from a gun are called _trajectories_.) If a plane flies in a direction always at the same angle from the north, how much farther will it travel than if it flew along the shortest path? (See Fig. 7.) (A path always at the same angle from the north is called a _loxodrome_, and a shortest path on a globe is called a _great circle_.) How should an engine be designed so that it will have the least vibration when it moves fast? In _physical problems_ like these, the answer is not a single number but a _formula_. What we want to do in any one of these problems is find a formula so that any one of the quantities may be calculated, given the behavior of the others. For example, here is a familiar problem in which the answer is a formula and not a number: [Illustration: FIG. 7. Paths of a flight.] [Illustration: FIG. 8. Room formulas.] How are the floor area of a room, its length, and its width related to each other? (See Fig. 8.) The answer is told in any one of three _equations_: (_floor area_) EQUALS (_length_) TIMES (_width_) (_length_) EQUALS (_floor area_) DIVIDED BY (_width_)