"nature makes them behave as they do according to the disposition of their organs; just as a clock, composed only of wheels and weights and springs can count the hours and measure the time more accurately than we can with all our intelligence. Descartes, R. (1641) Discourse on Methods, Fifth Part.
While Descartes' argument comes out of a cultural, scientific, linguistic, and religious environment quite remote from our own, it provides a high contrast background against which to evaluate human and animal abilities.
We can only evaluate this by evaluating individual organism's ability to adapt to novel environments, either experimentally, natural observation, or pure speculation!
Even this is not straightforward since a complex adaptive behavior may have components of different origins --even in members of the same species. (Human language may be an example of this, especially comparing early vs. later acquisition. See language notes.)
Finally, even this definition ignores the problems of specifying WHAT environments should be considered.
For now, it seems reasonable to take "intelligence" as characterized above.
This was experimentally demonstrated in chimpanzees, research which led to the use of pre-frontal lobotomy in the 1940-1960s as a psychiatric treatment for serious anxiety or other? disorders.
Passingham (1982) summarizes much of the research in the figure below, showing performance of various species on a two-choice visual discrimination task in which the organism picks one of two objects with a hidden reward under it. Essentially the subject must learn a rule --that the food is under a certain type of object despite other changes in position or features of the objects--and follow that rule in choosing.

Sketched from Fig. 5.8 in Passingham (1980).
Passingham concludes from these data that when mammals are ranked in terms of their improvement over a series of these problems, their rank is predicted by Jerison's (1973) measure of surplus brain cells.
Passingham is very cautious in assessing the data available to him and concludes the relationship between brain size and intelligence is very slight at best. Nevertheless, only a slight advantage (correlation) may have considerable evolutionary significance in explaining the rapid increase in hominid brains over the past 2-3 million years.
Greater intelligence would be particular important in this period of considerable climatic variation --variation that would place a premium on intelligence. (See Calvin's paper on intelligence.)
"When we say intelligent rather than clever we are often implying a substantial amount of looking ahead. " Calvin p.23
Using new methods, four recent studies of this relationship for the first time obtained estimates of brain size from high quality magnetic resonance imaging (MRI), instead of using external cranial dimensions. All four studies show about twice the correlation with intelligence than does head size. Willerman et al. (1991) report an estimated correlation of r = .35 (N = 40); Andreasen et al. (1993) found a correlation of r= .38 (N = 67); Raz et al (in press) found a correlation of r = .43 (N = 29); and Wickett et al. (in press) report a correlation of r = .395 (N = 40, all females). These are all statistically significant.
It appears that there is a small but reliable relationship between intelligence and brain size. This small difference has no importance in assessing individual intelligence --considering all the other factors involved. It does have evolutionary significance since a very small fitness advantage of a characteristic can, over generations, can have a dramatic effect --like the tripling of the hominid brain over the last 2 million years.
Hamilton (1935) reported that rats selected for 12 generations to be either "maze-bright" or "maze-dull" differed by about 2.5 standard deviations in brain weight. Within unselected control rats there was a correlation of r = .25 between maze ability and brain weight. Recently Anderson (1993) reported data on rats in which several cognitive tasks were given and a general factor extracted, and brain weights were obtained. The correlation between this general factor and brain weights in these rats was r = .48.
These animals studies only support the general idea that brain size is related to certain abilities. However keep in mind what Darwin observed, that the early experiences themselves can influence brain size. Indeed, it is conceivable that some increments in brain size occur from the extra post-natal experience and nutrition that the "premature" human brain gets as an indirect consequence of its neotenous condition.
Passingham notes that relative brain size predicts well the responsiveness ("curiosity") of zoo animals to novel objects over time. Suppose animals just get bored after exhausting their repertoire of responses to the object. Perhaps brain size indirectly reflects the variety of responses available to various species. Primates have the obvious handling advantage over carnivores, for example. And humans have a linguistic repertoire far exceeding that of any other primate.

Responsive of zoo animals to novel objects
Sketched after Fig. 5.10, Passingham (1980)
In phylogeny of a species, it drives evolution of the necessary mental capacities for obtaining food as well as in ontogeny of an individual, nutrition plays a complex role in maximizing the intelligence of the organism. (See diagram below adapted from Brown and Pollitt (1996) on the complex effects of malnutrition.)

In the United States, testing was connected with the rise of eugenics ("good genes") movement that advocated sterilization for mentally retarded and convicted criminals.
Binet in the early 1900s in France developed a test to evaluate children's potential for schooling. It assessed the basic skills of children that schoolteachers believed necessary for further learning. Basic number skills, language abilities, and logical assessment were prominent. From this developed the idea of group tests, norms, and mental age. Piaget, trained as a biologist, worked in Binet's lab and began to apply his ideas on biological epistemology (growth of knowledge) to humans.
Many in England, e.g. Spearman, Fisher, worked on the mathematical development of testing that we have today. Spearman was responsible for the idea of general intelligence as a "common factor--g--" to all performance--not seen directly but revealed in all adaptive behavior. The technique of "factor analysis" was devised to extract the common "factor" in a range of tests. Some speculated "g" reflected some general brain capacity such as size or speed.
In the USA, Binet's test was translated and revised for use here by psychologists at Stanford--resulting in the widely used Stanford-Binet test by L. Terman. R. M. Yerkes and other also devised a version of the Binet, "A Point Scale for Measuring Mental Ability, 1916" These tests were used in mobilzation for WWI--with minimal success. Nevertheless, they became important factors in clinical, school, and industrial settings.
While scores on these tests may predict success in schools, many critics point out that language skills and cultural biases make inferences about heredity on the basis of these tests very risky.
"I propose to show in this book that a man's natural abilities are derived by inheritance, under exactly the same limitations as are the form and physical features of the whole organic world. Consequently, as it is is easy, notwithstanding those limitations, to obtain by careful selection a permanent breed of dogs or horses gifted with peculiar powers of running, or doing anything else, so it would be quite practicable to produce a highly gifted race of men by judicious marriages during several consecutive marriages."
Galton, F. (1869). Hereditary genius, an inquiry into its laws and consequences .
PROCESSES PRODUCTS ENVIRONMENTAL
CONTRIBUTION
form genotype at regulatory genes amino acids, etc.
conception structural genes
enzyme production enyzmes nutrients (proteins,
carbohydrates, fats...)
regulated biochemical cell development and nutrients including
reactions metabolism oxygen etc prenatally
via the placenta (etc
here can have negative
effects, e.g. viruses,
inadequate oxygen,
hormone-like chemicals)
organized development of physiological structures nutrients and
body incl. c.n.s., including neural sensory-motor feedback
endocrine systems, etc. circuits and sense effects on neural
receptors development esp. late
fetal and early neonatal
periods
c.n.s commands, sensory Behavior sensory input nutrients
input
The most complex example is probably speech, where Darley, F. L., Aronson, A. E., & Brown, J. R. (1975). Motor speech disorders . Philadelphia: Saunders estimate 14,000 muscles must be controlled and at a very rapid rate!
Thus brain size may be a function of specific regulatory genes, it is also a function of nutrition and experience -especially early experiences. Natural selection can operate directly only on the regulatory gene component of the phenotype brain, though other genes relating to parenting and accelerated gestation--relevant to nutrition and neonatal experiences-- may also be enhanced.
(Recall the sketch on multiple effects of genes on the phenotype in my evolution notes. Plug the above variables into that sketch. You also might want to read Calvin's recent Scientific American article (Oct., 1994) on "The emergence of intelligence.")