By Joseph V. Collins
Excerpt from An simple Exposition of Grassmann's Ausdehnungslehre, or conception of Extension
The sum qf any variety of vectors is located through becoming a member of the start aspect of the second one vector to the top aspect of the 1st, the start aspect of the 3rd to the tip aspect of the second one. etc; the vector from the start element of the 1st vector to the tip element of the final is the sum required.
The sum and distinction of 2 vectors are the diagonals of the parallelogram whose adjoining aspects are the given vectors.
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Extra info for An elementary exposition of Grassmann's Ausdehnungslehre, or Theory of extension
By contrast, those whose predilections are more cheaply satisfied would receive smaller shares. Yet surely this would be to reward the prodigal unjustly. A more egregious example is that of a severely handicapped person who remains happy in spite of his infirmity. Somebody who is born or becomes disabled, and who had no opportunity to protect himself physically or financially (by buying insurance), is plainly entitled to special benefits whether or not he greets life with a smile. Egalitarian welfarism, however, says just the reverse.
Egalitarian welfarism cannot conceivably require that equality be attained by reducing everyone's welfare to the level of the most agonized and despondent individual. If one asserts, however, that the theory commands levelling up, not cutting the happy down to size, one needs another theory of justice to specify those rights that cannot be infringed in the course of establishing as nearly equal a distribution as possible. One might reply that egalitarian welfarism should be interpreted to require not that everybody's welfare be equalized, or made more nearly equal subject to certain constraints, but rather that resources should be ____________________ 24 I use the term "egalitarian welfarism" rather than the standard label "equality of welfare" because I consider, along with the view that everyone's welfare should be equalized, the view that their welfare should be increased to the same extent, even if 45 25 unequal welfare results.
Sidgwick is usually credited with the idea that utilitarianism is a working morality fit only for an elite, while the great sea of humanity must be tricked into virtue because they are insufficiently intelligent or self- disciplined or morally insightful to act on its dictates directly and reliably. See Sidgwick ( 1907), 490. Jonathan Glover lends qualified support to this view in Glover ( 1975), 188 -9, where he tries to meet Bernard Williams's critique of this argument in Smart and Williams ( 1973).