In the example at hand, the schema of visual reasoning involves at the start taking a number \(k\) to be represented by a column of \(k\) dots, thence taking the triangular array of \(n\) columns to represent the sum of the first \(n\) positive integers, thence taking that array combined with an inverted copy to make a rectangular array of \(n\) columns of \(n+1\) dots. For a schema starting this way to be universally applicable, it must be possible, given any positive integer \(n\), for the sum of the first \(n\) positive integers to be represented in the form of a triangular array, so that combined with an inverted copy one gets a rectangular array. This actually fails at the extreme case: \(n = 1\). The formula \((n.(n + 1))/2\) holds for this case; but that is something we know by substituting “1” for the variable in the formula, not by the visual method indicated. That method cannot be applied to \(n = 1\), because a single dot does not form a triangular array, and combined with a copy it does not form a rectangular array. But we can check that the method works for all positive integers after the first, using visual reasoning to assure ourselves that it works for 2 and that if the method works for \(k\) it works for \(k+1\). Together with this reflective thinking, the visual thinking sketched earlier constitutes following a proof of the formula for the \(n\)th triangular number for all integers \(n &gt; 1\), at least if the relaxed view of thinking through a proof is correct. Similar conclusions hold in the case of other “dot” arguments (Giaquinto 1993, 2007: ch. 8). So in some cases when the visual representation carries unwanted detail, the danger of over-generalisation in visual reasoning can be overcome.