To receive the "Methodical" achievement, you will need to reconstruct Walter Sickert's cane.
As Watson, travel to the Clinic and speak to the doctor. Ask him about Margie and the person that attacked her. Next, ask the doctor about the cane. Walk to the back of the clinic and inspect the large wooden case on the wall to open the cane puzzle.
The clues needed should be in a recent dialogue with the Artist (press "Y" to enter the sub-screen, and then select "Dialogues"), or choose the following pieces to assemble the cane (top to bottom):
Knob - 4th from left (fancy silver decorative one - round)
Shaft - 3rd from left (narrow black one)
Middle Section - 8th from left (fancy silver decorative one)
Tip - 4th from left (fancy silver decorative one - long)