Every negative quantity standing by itself is a mere creature of the mind and... those which are met with in calculations are only mere algebraical forms, incapable of representing any thing real and effective. ...That each of these algebraical forms being taken, with a proper consideration of its sign, is nothing else but the difference of two other absolute quantities, of which the greatest in the case of which the reasoning depends, is the least in the case in which the result of the calculation is to be applied. To say of a quantity that it becomes negative, is to employ an improper expression, and one leading... into error; and the true meaning of the expression is, that this absolute quantity does not belong to the system on which the reasonings have been established, but to another with which it is related; so that to apply the forms applicable to it for this first system, the sign + must be changed into the sign - in the preceding form. ...Nothing is to be changed... we have only to substitute a clear and true idea in the room of one faulty and useless; and this is the object of this work. I think that I have accomplished it by substituting for the notion of positive and negative quantities, which I have opposed, that of quantities which I call direct and inverse; and the geometry of position is that where the notion of positive and negative quantities standing by themselves is supplied by that of quantities direct and inverse. ...however these unintelligible forms ought not to be neglected, and... they may be employed like real forms, because they may be made to disappear by simple algebraical transformations; and there will then remain only explicit forms immediately applicable to the object proposed, provided, as it has been before observed, that these forms, conditions proposed, and suppositions on which the reasoning is grounded, are all consistent with each other.

What It Means

This is a placeholder explanation generated by the low-overhead model. It interprets "Every negative quantity standing by itself is a mere creature of the mind and... those which are met with in calculations are only mere algebraical forms, incapable of representing any thing real and effective. ...That each of these algebraical forms being taken, with a proper consideration of its sign, is nothing else but the difference of two other absolute quantities, of which the greatest in the case of which the reasoning depends, is the least in the case in which the result of the calculation is to be applied. To say of a quantity that it becomes negative, is to employ an improper expression, and one leading... into error; and the true meaning of the expression is, that this absolute quantity does not belong to the system on which the reasonings have been established, but to another with which it is related; so that to apply the forms applicable to it for this first system, the sign + must be changed into the sign - in the preceding form. ...Nothing is to be changed... we have only to substitute a clear and true idea in the room of one faulty and useless; and this is the object of this work. I think that I have accomplished it by substituting for the notion of positive and negative quantities, which I have opposed, that of quantities which I call direct and inverse; and the geometry of position is that where the notion of positive and negative quantities standing by themselves is supplied by that of quantities direct and inverse. ...however these unintelligible forms ought not to be neglected, and... they may be employed like real forms, because they may be made to disappear by simple algebraical transformations; and there will then remain only explicit forms immediately applicable to the object proposed, provided, as it has been before observed, that these forms, conditions proposed, and suppositions on which the reasoning is grounded, are all consistent with each other." in the context of "Lazare Carnot, Geometrie de Position (1803) as quoted by Boosey, "Carnot's Geometrie de Position," The Annual Review and History of Literature, Vol. 2 (1804) ed., Arthur Aikin, pp. 821-823." to mean that wisdom is timeless.