Though we cannot give rules which will be of much service when we have thus to divine the general form of the relation by which phenomena are connected, there are certain methods by which, in a narrower field, our investigations may be materially promoted;--certain special methods of obtaining laws from Observations. Of these we shall now proceed to treat. {{202}} CHAPTER VII. SPECIAL METHODS OF INDUCTION APPLICABLE TO QUANTITY. APHORISM XLIII. _There are special Methods of Induction applicable to Quantity; of which the principal are, the_ Method of Curves, _the_ Method of Means, _the_ Method of Least Squares, _and the_ Method of Residues. APHORISM XLIV. The Method of Curves _consists in drawing a curve of which the observed quantities are the Ordinates, the quantity on which the change of these quantities depends being the Abscissa. The efficacy of this Method depends upon the faculty which the eye possesses, of readily detecting regularity and irregularity in forms. The Method may be used to detect the Laws which the observed quantities follow: and also, when the Observations are inexact, it may be used to correct these Observations, so as to obtain data more true than the observed facts themselves._ APHORISM XLV. The Method of Means _gets rid of irregularities by taking the arithmetical mean of a great number of observed quantities. Its efficacy depends upon this; that in cases in which observed quantities are affected by other inequalities, besides that of which we wish to determine the law, the excesses_ above _and defects_ below _the quantities which the law in question would produce, will, in a collection of_ many _observations_, balance _each other._ {203} APHORISM XLVI. The Method of Least Squares _is a Method of Means, in which the mean is taken according to the condition, that the sum of the squares of the errours of observation shall be the least possible which the law of the facts allows. It appears, by the Doctrine of Chances, that this is the_ most probable _mean._ APHORISM XLVII. The Method of Residues _consists in subtracting, from the quantities given by Observation, the quantity given by any Law already discovered; and then examining the remainder, or_ Residue, _in order to discover the leading Law which it follows. When this second Law has been discovered, the quantity given by it may be subtracted from the first Residue; thus giving a_ Second Residue, _which may be examined in the same manner; and so on. The efficacy of this method depends principally upon the circumstance of the Laws of variation being successively smaller and smaller in amount (or at least in their mean effect); so that the ulterior undiscovered Laws do not prevent the Law in question from being_ prominent _in the observations._ APHORISM XLVIII.